solving problems with volume formulas

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Volume Problem Solving

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To solve problems on this page, you should be familiar with the following: Volume - Cuboid Volume - Sphere Volume - Cylinder Volume - Pyramid

This wiki includes several problems motivated to enhance problem-solving skills. Before getting started, recall the following formulas:

• Volume of sphere with radius \(r:\) \( \frac43 \pi r^3 \)
• Volume of cube with side length \(L:\) \( L^3 \)
• Volume of cone with radius \(r\) and height \(h:\) \( \frac13\pi r^2h \)
• Volume of cylinder with radius \(r\) and height \(h:\) \( \pi r^2h\)
• Volume of a cuboid with length \(l\), breadth \(b\), and height \(h:\) \(lbh\)

Volume Problem Solving - Basic

Volume - problem solving - intermediate, volume problem solving - advanced.

This section revolves around the basic understanding of volume and using the formulas for finding the volume. A couple of examples are followed by several problems to try.

Find the volume of a cube of side length \(10\text{ cm}\). \[\begin{align} (\text {Volume of a cube}) & = {(\text {Side length}})^{3}\\ & = {10}^{3}\\ & = 1000 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

Find the volume of a cuboid of length \(10\text{ cm}\), breadth \(8\text{ cm}\). and height \(6\text{ cm}\). \[\begin{align} (\text {Area of a cuboid}) & = l × b × h\\ & = 10 × 8 × 6\\ & = 480 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

I made a large ice cream cone of a composite shape of a cone and a hemisphere. If the height of the cone is 10 and the diameter of both the cone and the hemisphere is 6, what is the volume of this ice cream cone? The volume of the composite figure is the sum of the volume of the cone and the volume of the hemisphere. Recall the formulas for the following two volumes: \( V_{\text{cone}} = \frac13 \pi r^2 h\) and \( V_{\text{sphere}} =\frac43 \pi r^3 \). Since the volume of a hemisphere is half the volume of a a sphere of the same radius, the total volume for this problem is \[\frac13 \pi r^2 h + \frac12 \cdot \frac43 \pi r^3. \] With height \(h =10\), and diameter \(d = 6\) or radius \(r = \frac d2 = 3 \), the total volume is \(48\pi. \ _\square \)

Find the volume of a cone having slant height \(17\text{ cm}\) and radius of the base \(15\text{ cm}\). Let \(h\) denote the height of the cone, then \[\begin{align} (\text{slant height}) &=\sqrt {h^2 + r^2}\\ 17&= \sqrt {h^2 + 15^2}\\ 289&= h^2 + 225\\ h^2&=64\\ h& = 8. \end{align}\] Since the formula for the volume of a cone is \(\dfrac {1}{3} ×\pi ×r^2×h\), the volume of the cone is \[ \frac {1}{3}×3.14× 225 × 8= 1884 ~\big(\text{cm}^{2}\big). \ _\square\]

Find the volume of the following figure which depicts a cone and an hemisphere, up to \(2\) decimal places. In this figure, the shape of the base of the cone is circular and the whole flat part of the hemisphere exactly coincides with the base of the cone (in other words, the base of the cone and the flat part of the hemisphere are the same). Use \(\pi=\frac{22}{7}.\) \[\begin{align} (\text{Volume of cone}) & = \dfrac {1}{3} \pi r^2 h\\ & = \dfrac {1 × 22 × 36 × 8}{3 × 7}\\ & = \dfrac {6336}{21} = 301.71 \\\\ (\text{Volume of hemisphere}) & = \dfrac {2}{3} \pi r^3\\ & = \dfrac {2 × 22 × 216}{3 × 7}\\ & = \dfrac {9504}{21} = 452.57 \\\\ (\text{Total volume of figure}) & = (301.71 + 452.57) \\ & = 754.28.\ _\square \end{align} \]

Try the following problems.

Find the volume (in \(\text{cm}^3\)) of a cube of side length \(5\text{ cm} \).

A spherical balloon is inflated until its volume becomes 27 times its original volume. Which of the following is true?

Bob has a pipe with a diameter of \(\frac { 6 }{ \sqrt { \pi } }\text{ cm} \) and a length of \(3\text{ m}\). How much water could be in this pipe at any one time, in \(\text{cm}^3?\)

What is the volume of the octahedron inside this \(8 \text{ in}^3\) cube?

A sector with radius \(10\text{ cm}\) and central angle \(45^\circ\) is to be made into a right circular cone. Find the volume of the cone.

\[\] Details and Assumptions:

• The arc length of the sector is equal to the circumference of the base of the cone.

Three identical tanks are shown above. The spheres in a given tank are the same size and packed wall-to-wall. If the tanks are filled to the top with water, then which tank would contain the most water?

A chocolate shop sells its products in 3 different shapes: a cylindrical bar, a spherical ball, and a cone. These 3 shapes are of the same height and radius, as shown in the picture. Which of these choices would give you the most chocolate?

\[\text{ I. A full cylindrical bar } \hspace{.4cm} \text{ or } \hspace{.45cm} \text{ II. A ball plus a cone }\]

How many cubes measuring 2 units on one side must be added to a cube measuring 8 units on one side to form a cube measuring 12 units on one side?

This section involves a deeper understanding of volume and the formulas to find the volume. Here are a couple of worked out examples followed by several "Try It Yourself" problems:

\(12\) spheres of the same size are made from melting a solid cylinder of \(16\text{ cm}\) diameter and \(2\text{ cm}\) height. Find the diameter of each sphere. Use \(\pi=\frac{22}{7}.\) The volume of the cylinder is \[\pi× r^2 × h = \frac {22×8^2×2}{7}= \frac {2816}{7}.\] Let the radius of each sphere be \(r\text{ cm}.\) Then the volume of each sphere in \(\text{cm}^3\) is \[\dfrac {4×22×r^3}{3×7} = \dfrac{88×r^3}{21}.\] Since the number of spheres is \(\frac {\text{Volume of cylinder}}{\text {Volume of 1 sphere}},\) \[\begin{align} 12 &= \dfrac{2816×21}{7×88×r^3}\\ &= \dfrac {96}{r^3}\\ r^3 &= \dfrac {96}{12}\\ &= 8\\ \Rightarrow r &= 2. \end{align}\] Therefore, the diameter of each sphere is \[2\times r = 2\times 2 = 4 ~(\text{cm}). \ _\square\]

Find the volume of a hemispherical shell whose outer radius is \(7\text{ cm}\) and inner radius is \(3\text{ cm}\), up to \(2\) decimal places. We have \[\begin{align} (\text {Volume of inner hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × R^3\\ & = \dfrac {1 × 4 × 22 × 27}{2 × 3 × 7}\\ & = \dfrac {396}{7}\\ & = 56.57 ~\big(\text{cm}^{3}\big) \\\\ (\text {Volume of outer hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × r^3\\ & = \dfrac {1 × 4 × 22 × 343}{2 × 3 × 7}\\ & = \dfrac {2156}{7}\\ & = 718.66 ~\big(\text{cm}^{3}\big) \\\\ (\text{Volume of hemispherical shell}) & = (\text{V. of outer hemisphere}) - (\text{V. of inner hemisphere})\\ & = 718.66 - 56.57 \\ & = 662.09 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

A student did an experiment using a cone, a sphere, and a cylinder each having the same radius and height. He started with the cylinder full of liquid and then poured it into the cone until the cone was full. Then, he began pouring the remaining liquid from the cylinder into the sphere. What was the result which he observed?

There are two identical right circular cones each of height \(2\text{ cm}.\) They are placed vertically, with their apex pointing downwards, and one cone is vertically above the other. At the start, the upper cone is full of water and the lower cone is empty.

Water drips down through a hole in the apex of the upper cone into the lower cone. When the height of water in the upper cone is \(1\text{ cm},\) what is the height of water in the lower cone (in \(\text{cm}\))?

On each face of a cuboid, the sum of its perimeter and its area is written. The numbers recorded this way are 16, 24, and 31, each written on a pair of opposite sides of the cuboid. The volume of the cuboid lies between \(\text{__________}.\)

A cube rests inside a sphere such that each vertex touches the sphere. The radius of the sphere is \(6 \text{ cm}.\) Determine the volume of the cube.

If the volume of the cube can be expressed in the form of \(a\sqrt{3} \text{ cm}^{3}\), find the value of \(a\).

A sphere has volume \(x \text{ m}^3 \) and surface area \(x \text{ m}^2 \). Keeping its diameter as body diagonal, a cube is made which has volume \(a \text{ m}^3 \) and surface area \(b \text{ m}^2 \). What is the ratio \(a:b?\)

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction \( \frac{m}{n} \), for relatively prime integers \(m\) and \(n\). Compute \(m+n\).

The square-based pyramid A is inscribed within a cube while the tetrahedral pyramid B has its sides equal to the square's diagonal (red) as shown.

Which pyramid has more volume?

Please remember this section contains highly advanced problems of volume. Here it goes:

Cube \(ABCDEFGH\), labeled as shown above, has edge length \(1\) and is cut by a plane passing through vertex \(D\) and the midpoints \(M\) and \(N\) of \(\overline{AB}\) and \(\overline{CG}\) respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

If the American NFL regulation football

has a tip-to-tip length of \(11\) inches and a largest round circumference of \(22\) in the middle, then the volume of the American football is \(\text{____________}.\)

Note: The American NFL regulation football is not an ellipsoid. The long cross-section consists of two circular arcs meeting at the tips. Don't use the volume formula for an ellipsoid.

Answer is in cubic inches.

Consider a solid formed by the intersection of three orthogonal cylinders, each of diameter \( D = 10 \).

What is the volume of this solid?

Consider a tetrahedron with side lengths \(2, 3, 3, 4, 5, 5\). The largest possible volume of this tetrahedron has the form \( \frac {a \sqrt{b}}{c}\), where \(b\) is an integer that's not divisible by the square of any prime, \(a\) and \(c\) are positive, coprime integers. What is the value of \(a+b+c\)?

Let there be a solid characterized by the equation \[{ \left( \frac { x }{ a } \right) }^{ 2.5 }+{ \left( \frac { y }{ b } \right) }^{ 2.5 } + { \left( \frac { z }{ c } \right) }^{ 2.5 }<1.\]

Calculate the volume of this solid if \(a = b =2\) and \(c = 3\).

• Surface Area

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Volume Worksheets

This humongous collection of printable volume worksheets is sure to walk middle and high school students step-by-step through a variety of exercises beginning with counting cubes, moving on to finding the volume of solid shapes such as cubes, cones, rectangular and triangular prisms and pyramids, cylinders, spheres and hemispheres, L-blocks, and mixed shapes. Brimming with learning and backed by application the PDFs offer varied levels of difficulty.

List of Volume Worksheets

Counting Cubes

• Volume of Cubes
• Volume of Rectangular Prisms
• Volume of Triangular Prisms

Volume of Mixed Prisms

• Volume of Cones
• Volume of Cylinders

Volume of Spheres and Hemispheres

• Volume of Rectangular Pyramids
• Volume of Triangular Pyramids

Volume of Mixed Pyramids

Volume of Mixed Shapes

Volume of Composite Shapes

Explore the Volume Worksheets in Detail

Work on the skill of finding volume with this batch of counting cubes worksheets. Count unit cubes to determine the volume of rectangular prisms and solid blocks, draw prisms on isometric dot paper and much more.

Volume of a Cube

Augment practice with this unit of pdf worksheets on finding the volume of a cube comprising problems presented as shapes and in the word format with side length measures involving integers, decimals and fractions.

Volume of a Rectangular Prism

This batch of volume worksheets provides a great way to learn and perfect skills in finding the volume of rectangular prisms with dimensions expressed in varied forms, find the volume of L-blocks, missing measure and more.

Volume of a Triangular Prism

Encourage students to work out the entire collection of printable worksheets on computing the volume of triangular prism using the area of the cross-section or the base and leg measures and practice unit conversions too.

Navigate through this collection of volume of mixed prism worksheets featuring triangular, rectangular, trapezoidal and polygonal prisms. Bolster practice with easy and moderate levels classified based on the number range used.

Volume of a Cone

Motivate learners to use the volume of a cone formula efficiently in the easy level, find the radius in the moderate level and convert units in the difficult level, solve for volume using slant height, and find the volume of a conical frustum too.

Volume of a Cylinder

Access our volume of a cylinder worksheets to practice finding the radius from diameter, finding the volume of cylinders with parameters in integers and decimals, find the missing parameters, solve word problems and more!

Take the hassle out of finding the volume of spheres and hemispheres with this compilation of pdf worksheets. Gain immense practice with a wide range of exercises involving integers and decimals.

Volume of a Rectangular Pyramid

This exercise is bound to help learners work on the skill of finding the volume of rectangular pyramids with dimensions expressed as integers, decimals and fractions in easy and moderate levels.

Volume of a Triangular Pyramid

Help children further their practice with this bundle of pdf worksheets on determining the volume of triangular pyramids using the measures of the base area or height and base. The problems are offered as 3D shapes and in word format in varied levels of difficulty.

Gain ample practice in finding the volume of pyramids with triangular, rectangular and polygonal base faces presented in two levels of difficulty. Apply relevant formulas to find the volume using the base area or the other dimensions provided.

Upscale practice with an enormous collection of printable worksheets on finding the volume of solid shapes like prisms, cylinders, cones, pyramids and revision exercises to revisit concepts with ease.

Learn to find the volume of composite shapes that are a combination of two or more solid 3D shapes. Begin with counting squares, find the volume of L -blocks, and compound shapes by adding or subtracting volumes of decomposed shapes.

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Area of polygons

Volume formula

Here you will learn about volume formulas, including the formula for calculating the volume of 3D shapes and how to use the volume formula to solve problems.

Students will first learn about the volume formula as part of measurement and data in the 5 th grade, and continue to expand on their knowledge in geometry in the 6 th and 7 th grade.

What is a volume formula?

A volume formula is a formula used to calculate the volume of a 3D shape.

Volume is the amount of space there is inside a shape.

To calculate the volume of an object in three dimensions, you can use the various volume formulas.

Volume is measured in cubic units.

• ft^3- cubic feet
• \mathrm{in}^3- cubic inches
• \mathrm{cm}^3 - cubic centimeters
• m^3 - cubic meters
• \mathrm{mm}^3 - cubic millimeters

Volume can also be described using units of capacity such as milliliters, liters, pints or gallons.

Common Core State Standards

How does this relate to 5 th, 6 th and 7 th grade math?

• Grade 5: Measurement and Data (5.MD.C.5b) Apply the formulas V = l \times w \times h and V = b \times h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.
• Grade 6: Geometry (6.G.A.2) Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l \times w \times h and V = b \times h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
• Grade 7: Geometry (7.G.B.6) Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
• Grade 8: Geometry (8.G.C.9) Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

How to use volume formula

In order to use volume formula:

Write down the formula.

Substitute the values into the formula.

Calculate the volume of the shape.

Write the answer, including the units.

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Volume formula examples

Example 1: volume of a cylinder.

Calculate the volume of the cylinder below.

The 3D shape is a cylinder. The formula you need to use is \text{Volume}=\pi r^2h.

2 Substitute the values into the formula.

Calculate the area of a circle (the circular base) and then multiply it by the height of the shape.

The values you need to substitute into the formula are the radius of the base (r=5) and the height of the shape (h=13).

\begin{aligned} \text{Volume}&=\pi r^2h \\\\ &=\pi \times 5^2 \times 13 \end{aligned}

3 Calculate the volume of the shape.

V=325 \, \pi=1021.017…

4 Write the answer, including the units.

The dimensions of the cylinder were given in centimeters, so the volume (the amount of three-dimensional space inside the shape) will be in cubic centimeters (cm^3).

V=1021 \ cm^3 (to the nearest integer)

Example 2: volume of a sphere

Calculate the volume of a sphere with a diameter of 9 \, mm.

The 3D shape is a sphere. The formula you need to use is \text{Volume}=\cfrac{4}{3} \, \pi r^3.

You need to substitute the value of the radius of the sphere into the formula. You need to divide the diameter by 2 to calculate the radius r.

r=9\div 2=4.5

\begin{aligned} \text{Volume}&=\cfrac{4}{3} \, \pi r^3 \\\\ &=\cfrac {4}{3}\times \pi \times 4.5^3\end{aligned}

V=\cfrac{243}{2} \, \pi=381.703…

The dimensions of the sphere were given in millimeters, so the volume of the sphere will be in cubic millimeters (mm^3).

V=382 \ mm^3 (to the nearest integer)

Example 3: missing dimension

The volume of the rectangular prism is 2,016 \mathrm{~cm}^3.

Calculate the value of x.

The 3D shape is a rectangular prism. The formula you need to use is

\text{Volume}=l \times w \times h.

The length (l), the width (w) and the height (h) of the rectangular prism are interchangeable, so it doesn’t matter which dimension l or w or h.

The values you need to substitute into the formula are \text{Volume}=2,016, l=8 and w=14.

For the height (h) you can use x.

\begin{aligned} \text{Volume}&=l\times w\times h \\\\ 2,016&=8 \times 14\times x \\\\ 2,016&=112\times x \end{aligned}

In this case, you already know the volume. You divide the volume by the area of the base to find the missing height.

x=2,016\div 112=18.

The dimensions of the rectangular prism were given in centimeters and the volume was given in cubic centimeters (cm^3).

Since x is a dimension of the rectangular prism, it is recorded in just units, not units cubed.

Example 4: missing dimension

The volume of the square pyramid is 66,000 \mathrm{~cm}^3.

Calculate the height of the pyramid.

The 3D shape is a pyramid. The formula you need to use is,

\text{Volume}=\cfrac{1}{3} \times \text{Area of base} \times \text{Height}.

First, you need to calculate the area of the square base: 60^2=60\times 60=3,600.

The values you need to substitute into the formula are \text{Volume}=66, 000 and \text{Area of base}=3,600.

For the height, you can use h.

\begin{aligned} \text{Volume}&=\cfrac{1}{3}\times \text{Area of base} \times \text{Height}\\\\ 66,000&=\cfrac {1}{3}\times 3,600\times h\\\\ 66,000&=1,200\times h \end{aligned}

h=66,000\div 1,200=55.

The dimensions of the pyramid were given in centimeters and the volume was given in cubic centimeters (cm^3).

Since h is a dimension of the square pyramid, it is recorded in just units, not units cubed.

Example 5: volume of a compound shape

This shape is made from a rectangular prism and a square based pyramid. Calculate the volume of the 3D shape.

The 3D shape is made from a rectangular prism and a pyramid. Find the volume of each piece and then add the two volumes together. The formulas you need to use are:

The values you need to substitute into the formulae are

The volume of the shape is:

\text{Total Volume}=2,160+384=2,544.

The dimensions of the 3D shape were given in centimeters, so the volume will be in cubic centimeters (cm^3).

V=2,544 \, cm^3

Example 6: volume of a compound shape

This shape is made from a cylinder and a cone. Calculate the volume of the 3D shape.

The 3D shape is made from a cylinder and a cone. You can find the volume of each piece and then add the two volumes together. The formulas you need to use are:

You need to divide the diameter by 2 to calculate the radius. The values you need to substitute into the formula are:

\text{Total Volume}=\cfrac{175}{4} \, \pi+\cfrac{75}{4} \, \pi=\cfrac{125}{2} \, \pi.

V=\cfrac{125}{2} \, \pi\ cm^3 (in terms of \pi)

196.3 \, cm^3 (to 1 decimal place)

Teaching tips for the volume formula

• Conduct interactive demonstrations to illustrate how volume formulas work. Find the volume of water or sand using different containers of varying shapes and sizes, by asking students to estimate and calculate the volume.
• Offer a wide range of exercises and problems for students to practice calculating volume. Provide different shapes and sizes to challenge them and encourage critical thinking.
• Regularly assess students’ understanding of volume formulas through quizzes, worksheets, or project assignments. Provide timely and constructive feedback to address any misconceptions or errors.

Easy mistakes to make

• Forgetting to use units, or using the incorrect units You should always include units in your answer. Volume is measured in cubic units (for example , mm^3, cm^3, m^3 ).
• Calculating with different units You need to make sure all measurements are in the same units before calculating the volume. (For example, you can’t have some measurements in centimeters and some in meters).
• Using the wrong formula when calculating volume Be careful to make sure you use the correct formula for the volume of the three-dimensional shape. Volume formulas can be easily confused with each other and with formulas for calculating the surface area of a three-dimensional object.

Related volume lessons

• Volume of a prism
• Volume of a rectangular prism
• Volume of a cube
• Volume of a pyramid

Practice volume formula questions

1. Calculate the volume.

The volume formula to calculate the volume of the cylinder is \text{Volume} =\pi r^2h.

You need to substitute in the values r=5 and h=6.

\begin{aligned}\text{Volume} &=\pi r^2h \\\\ &= \pi \times 5^2 \times 6 \\\\ &=150 \, \pi \\\\ &=471.23…\end{aligned}

The volume of the cylinder is 471 \, cm^3.

2. Calculate the volume of this pyramid.

The volume formula to calculate the volume of the pyramid is \text{Volume} =\cfrac{1}{3}\times \text{Area of base}\times \text{Height}.

You need to find the area of the base of the pyramid.

A=5\times 6=30

\begin{aligned}\text{Volume} &=\cfrac{1}{3}\times \text{area of base}\times \text{height} \\\\ &= \cfrac{1}{3}\times 30\times 8 \\\\ &=80\end{aligned}

The volume of the pyramid is 80 \, cm^3.

3. The volume of this rectangular prism is 720 \, mm^3. Calculate the length of the missing side.

The volume formula to calculate the volume of the rectangular prism is \text{Volume} =l\times w\times h.

You need to substitute in the values given, using x for the unknown length, into the volume formula.

\begin{aligned}\text{Volume} &=l\times w\times h\\\\ 720&=x\times 5\times 9\\\\ 720&=x\times 45\end{aligned}

The missing length is x=720\div 45=16.

The missing length of the rectangular prism is 16 \, mm.

4. The volume of a cylinder is 160 \, \pi \, cm^3. The height of the cylinder is 6.4 \, cm. Find the radius of the cylinder.

You need to substitute in the values given into the formula:

\begin{aligned}\text{Volume} &=\pi r^2h.\\\\ 160 \, \pi&=\pi \times r^2\times 6.4 \end{aligned}

Solve for the missing length:

\begin{aligned}r^2&=\cfrac{160 \, \pi}{\pi \times 6.4}\\\\ r^2&=25 \; \text{(think about what number times itself is 25.}) \\\\ r&=5, \text{because} \; 5 \times 5 = 25. \end{aligned}

The radius of the cylinder is 5 \, cm.

5. Calculate the volume of the shape below.

The volume of the rectangular prism is:

\begin{aligned}\text{Volume}&=l\times w\times h\\\\ &=10\times 10\times 10\\\\ &=1,000\end{aligned}

Or you can take one side of the cube and cube it.

The volume of the pyramid is

\begin{aligned}\text{Volume}&=\cfrac{1}{3}\times \text{area of base}\times \text{height}\\\\ &=\cfrac{1}{3}\times 10\times 10\times 6\\\\ &=200\end{aligned}

The total volume can be found by adding the two volumes together.

\text{Total Volume}=1,000+200=1,200 \, cm^3

6. Calculate the volume of the shape below.

The volume of the cylinder is:

\begin{aligned}\text{Volume}&=\pi r^2h\\\\ &=\pi \times 6^2\times 12\\\\ &=432 \, \pi \end{aligned}

The volume of the cone is:

\begin{aligned}\text{Volume}&=\cfrac{1}{3} \, \pi r^2h\\\\ &=\cfrac{1}{3}\times \pi \times 6^2\times 7\\\\ &=84 \, \pi\end{aligned}

\text{Total Volume}=432 \, \pi+84 \, \pi=516 \, \pi \, cm^3

Volume formula FAQs

The formula for calculating the volume of a triangular prism is V=\cfrac{1}{2} \, B \times h , where "V" represents the volume, "B" represents the area of the triangular base, and "h" represents the height of the prism.

No, cuboid is another word for a rectangular prism.

The base is always the two congruent opposite polygons, not just the bottom. The bases are connected by the lateral faces, which is why they are the height, even if the prism is laying on its side.

The next lessons are

• Surface area

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Volume formula

The volume of a 3D shape or geometric figure is the amount of space it contains. Volume is well-defined for many common shapes; the formulas for some common shapes are shown below.

The volume, V, of a cube with edge, s, is:

The volume, V, of a prism is:

where B is area of the base and h is the height of the prism.

Rectangular prism

The volume, V, of a rectangular prism is:

where l is the length, w is the width, and h is the height of the rectangular prism.

The volume, V, of a pyramid is:

where B is area of the base and h is the height of the pyramid.

The volume, V, of a cone is:

where r is radius of the base and h is the height of the cone.

The volume, V, of a cylinder is:

V = πr 2 h

where r is the radius of the base and h is the height of the cylinder.

The volume, V, of a sphere with radius, r, is:

Find the volume of the rectangular prism below.

The volume of the rectangular prism is:

V = 5 × 3 × 2 = 30 cm 3

Volume of a composite figure

3D composite figures are figures that are made up of two or more types of figures. Their volumes can be calculated by breaking them down into their components, calculating the volumes of each component, then summing them to find the total volume of the composite figure.

The grain silo below is made up of a right cylinder and a right circular cone. Find the amount of grain, in cubic meters, the silo can hold when full.

The cylinder has a radius of 8 m and height of 10 m. The cone also has a radius of 8 m, since it sits on top of the cylinder, and its height is 5 m.

The volume of the cylinder is:

V cylinder = π × 8 2 × 10 = 640π m 3

The volume of the cone is:

The volume, V, of the silo is the sum of the volumes of the cylinder and cone:

The silo can hold 2345.72 cubic meters of grain.

Using a unit cube to find volume

One way to find the volume of a figure is to determine how many unit cubes it takes to fill the figure. A unit cube has side lengths of 1 and a volume of 1.

The rectangular prism below has a length of 5, width of 3, and height of 2.

You can evenly stack 2 layers of unit cubes, containing a total of 15 unit cubes each into the rectangular prism to find its volume of 30 unit cubes.

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HiSET: Math : Use volume formulas to solve problems

Study concepts, example questions & explanations for hiset: math, all hiset: math resources, example questions, example question #1 : use volume formulas to solve problems.

Step 1: Find the diameter.

If we are given the diameter, the length of the radius is one-half the diameter. 

Step 2: Recall the volume formula...

Example Question #161 : Hi Set: High School Equivalency Test: Math

The formula for the volume of a pyramid is 

Note, the area of the base of the pyramid is

Example Question #1 : Pyramids

A pyramid with a square base is inscribed inside a right cone with radius 24 and height 10.

Give the volume of the pyramid.

None of the other choices gives the correct response.

The circle has radius 24, so its diameter - and the lengths of the diagonals - is twice this, or 48.

The area of a square - which is also a rhombus - is equal to half the product of the lengths of its diagonals, so

The base of a right pyramid with height 6 is a regular hexagon with sides of length 6.

Give its volume.

The regular hexagonal base can be divided by its diameters into six equilateral triangles, as seen below:

the total area of the base is six times this.

The total area of the base is six times this, or

Example Question #4 : Pyramids

A right pyramid and a right rectangular prism both have square bases. The base of the pyramid has sides that are 20% longer than those of the bases of the prism; the height of the pyramid is 20% greater than that of the prism.

Which of the following is closest to being correct?

The volume of the pyramid is 61.6% less than that of the prism.

The volume of the pyramid is 74.4% less than that of the prism.

The volume of the pyramid is 33.3% less than that of the prism.

The volume of the pyramid is 42.4% less than that of the prism.

The volume of the pyramid is 82.9% less than that of the prism.

Example Question #5 : Pyramids

The pyramid in question can be seen in the diagram below:

Example Question #2 : Cones

Example Question #3 : Cones

A right square pyramid has height 10 and a base of perimeter 36.

Inscribe a right cone inside this pyramid. What is its volume?

Example Question #4 : Cones

A right square pyramid has height 10 and a base of area 36.

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SAT (Fall 2023)

Course: sat (fall 2023)   >   unit 6, volume word problems | lesson.

• Right triangle word problems | Lesson
• Congruence and similarity | Lesson
• Right triangle trigonometry | Lesson
• Angles, arc lengths, and trig functions | Lesson
• Circle theorems | Lesson
• Circle equations | Lesson
• Complex numbers | Lesson

What are volume word problems, and how frequently do they appear on the test?

• Calculate the volumes and dimensions of three-dimensional solids
• Determine how dimension changes affect volume

How do I calculate the volumes and dimensions of shapes?

Volume word problem: gold ring, volume of a cone, the volumes of three-dimensional solids.

• Find the volume formula for the solid.
• Plug the dimensions into the formula.
• Evaluate the volume.
• Plug the volume and any known dimensions into the formula.
• Isolate the unknown dimension.
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

How do changing dimensions affect volume?

How volume changes when dimensions change, impact of increasing the radius, the effect of changing dimensions on volume.

• (Choice A)   A cylinder with 2 ‍   times the radius and 1 2 ‍   the height of cylinder A ‍   A A cylinder with 2 ‍   times the radius and 1 2 ‍   the height of cylinder A ‍  
• (Choice B)   A cylinder with 2 ‍   times the radius and 1 4 ‍   the height of cylinder A ‍   B A cylinder with 2 ‍   times the radius and 1 4 ‍   the height of cylinder A ‍  
• (Choice C)   A cylinder with 1 2 ‍   the radius and 8 ‍   times the height of cylinder A ‍   C A cylinder with 1 2 ‍   the radius and 8 ‍   times the height of cylinder A ‍  
• (Choice D)   A cylinder with 1 3 ‍   the radius and 9 ‍   times the height of cylinder A ‍   D A cylinder with 1 3 ‍   the radius and 9 ‍   times the height of cylinder A ‍  
• (Choice A)   450 ‍   A 450 ‍  
• (Choice B)   900 ‍   B 900 ‍  
• (Choice C)   1,800 ‍   C 1,800 ‍  
• (Choice D)   3,600 ‍   D 3,600 ‍  

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How to Calculate Volume

Last Updated: April 25, 2023 Fact Checked

This article was co-authored by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. There are 12 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,458,368 times.

The volume of a shape is the measure of how much three-dimensional space that shape takes up. [1] X Research source You can also think of the volume of a shape as how much water (or air, or sand, etc.) the shape could hold if it was filled completely. Common units of volume include cubic centimeters (cm 3 ), cubic meters (m 3 ), cubic inches (in 3 ), and cubic feet (ft 3 ). [2] X Research source This article will teach you how to calculate the volume of six different three-dimensional shapes that are commonly found on math tests, including cubes, spheres, and cones. You might notice that a lot of the volume formulas share similarities that can make them easier to remember. See if you can spot them along the way!

Calculating the Volume of a Cube

• A 6-sided die is a good example of a cube you might find in your house. Sugar cubes, and children's letter blocks are also usually cubes.

• To find s 3 , simply multiply s by itself 3 times: s 3 = s * s * s

• If you are not 100% sure that your shape is a cube, measure each of the sides to determine if they are equal. If they are not, you will need to use the method below for Calculating the Volume of a Rectangular Solid.

Calculating the Volume of a Rectangular Prism

• A cube is really just a special rectangular solid in which the sides of all of the rectangles are equal.

• Example: The length of this rectangular solid is 4 inches, so l = 4 in.
• Don't worry too much about which side is the length, which is the width, etc. As long as you end up with three different measurements, the math will come out the same regardless of how your arrange the terms.

• Example: The width of this rectangular solid is 3 inches, so w = 3 in.
• If you are measuring the rectangular solid with a ruler or tape measure, remember to take and record all measurements in the same units. Don't measure one side in inches another in centimeters; all measurements must use the same unit!

• Example: The height of this rectangular solid is 6 inches, so h = 6 in.

• In our example, l = 4, w = 3, and h = 6. Therefore, V = 4 * 3 * 6, or 72.

• If the measurements of our rectangular solid were: length = 2 cm, width = 4 cm, and height = 8 cm, the Volume would be 2 cm * 4 cm * 8 cm, or 64cm 3 .

Calculating the Volume of a Cylinder

• A can is a good example of a cylinder, so is a AA or AAA battery.

• In some geometry problems the answer will be given in terms of pi, but in most cases it is sufficient to round pi to 3.14. Check with your instructor to find out what she would prefer.
• The formula for finding the volume of a cylinder is actually very similar to that for a rectangular solid: you are simply multiplying the height of the shape by the surface area of its base. In a rectangular solid, that surface area is l * w, for the cylinder it is πr 2 , the area of a circle with radius r.

• Another option is to measure the circumference of the cylinder (the distance around it) using a tape measure or a length of string that you can mark and then measure with a ruler. Then plug the measurement into the formula: C (circumference) = 2πr. Divide the circumference by 2π (6.28) and that will give you the radius.
• For example, if the circumference you measured was 8 inches, the radius would be 1.27in.
• If you need a really precise measurement, you might use both methods to make sure that your measurements are similar. If they are not, double check them. The circumference method will usually yield more accurate results.

• If the radius of the circle is equal to 4 inches, the area of the base will be A = π4 2 .
• 4 2 = 4 * 4, or 16. 16 * π (3.14) = 50.24 in 2
• If the diameter of the base is given instead of the radius, remember that d = 2r. You simply need to divide the diameter in half to find the radius.

• V = π4 2 10
• π4 2 = 50.24
• 50.24 * 10 = 502.4

Calculating the Volume of a Regular Square Pyramid

• We most commonly imagine a pyramid as having a square base, and sides that taper up to a single point, but the base of a pyramid can actually have 5, 6, or even 100 sides!
• A pyramid with a circular base is called a cone, which will be discussed in the next method.

• The volume formula is the same for right pyramids, in which the apex is directly above the center of the base, and for oblique pyramids, in which the apex is not centered.

• The formula for the area of a triangle is: A = 1/2bh, where b is the base of the triangle and h is the height.
• It is possible to find the area of any regular polygon using the formula A = 1/2pa, where A is the area, p is the perimeter of the shape, and a is the apothem, or distance from the center of the shape to the midpoint of any of its sides. This is a pretty involved calculation that goes beyond the scope of this article, but check out Calculate the Area of a Polygon for some great instructions on how to use it. Or you can make your life easy and search for a Regular Polygon Calculator online. [15] X Research source

• If we had a different pyramid, with a pentagonal base with area 26, and height of 8, the volume would be: 1/3 * 26 * 8 = 69.33.

Calculating the Volume of a Cone

• If the vertex of the cone is directly above the center of the circular base, the cone is called a "right cone". If it is not directly over the center, the cone is called an "oblique cone." Fortunately, the formula for calculating the area of a cone is the same whether it is right or oblique.

• The πr 2 part of the formula refers to the area of the circular base of the cone. The formula for the volume of the cone is thus 1/3bh, just like the formula for the volume of a pyramid in the method above!

• In the example in the diagram, the radius of the circular base of the cone is 3 inches. When we plug that into the formula we get: A = π3 2 .
• 3 2 = 3 *3, or 0, so A = 9π.
• A = 28.27in 2

• In our example, 141.35 * 1/3 = 47.12, the volume of our cone.
• To restate it, 1/3π3 2 5 = 47.12

Calculating the Volume of a Sphere

• For example, if you measure a ball and find its circumference is 18 inches, divide that number by 6.28 and you will find that the radius is 2.87in.
• Measuring a spherical object can be a little tricky, so you might want to take 3 different measurements, and then average them together (add the three measurements together, then divide by 3) to make sure you have the most accurate value possible.
• For example, if your three circumference measurements were 18 inches, 17.75 inches, and 18.2 inches, you would add those three values together (18 + 17.5 + 18.2 = 53.95) and divide that value by 3 (53.95/3 = 17.98). Use this average value in your volume calculations.

• In our example, 36 * 3.14 = 113.09.

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• ↑ https://www.nist.gov/pml/owm/si-units-volume
• ↑ http://www.mathsisfun.com/measure/us-standard-volume.html
• ↑ https://www.mathsisfun.com/definitions/cube.html
• ↑ Grace Imson, MA. Math Instructor, City College of San Francisco. Expert Interview. 1 November 2019.
• ↑ http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_3Dprisms.xml
• ↑ https://www.mathsisfun.com/definitions/cylinder.html
• ↑ http://www.mathwords.com/p/pyramid.htm
• ↑ http://www.mathwords.com/r/regular_pyramid.htm
• ↑ http://www.calculatorsoup.com/calculators/geometry-plane/polygon.php
• ↑ http://www.mathopenref.com/cone.html
• ↑ https://www.mathsisfun.com/definitions/sphere.html
• ↑ https://www.splashlearn.com/math-vocabulary/geometry/volume

About This Article

To calculate volume with a cube, use the formula v = s^3, where s is the length of the sides of the cube. To calculate the volume of a cylinder, use the formula v = hπr^2, where r is the radius of the base, h is the height, and π is pi. If you're trying to find the volume of a rectangular prism, use the formula v = lwh, where l is the length, w is the width, and h is the height. If you need to learn how to calculate the volume of a sphere or pyramid, keep reading the article! Did this summary help you? Yes No

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Volume Math Problems: Homework Help for Students

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Need help finding the volume of an object? This article explains how to find the volume of many 3-dimensional shapes, including rectangular prisms, cylinders and spheres.

Calculating Volume

Rectangular prisms.

A rectangular prism is a solid, 3-dimensional object with three sets of parallel sides that are perpendicular to one another. A pack of playing cards is an example of a rectangular prism and so is a paperback book.

To find the volume of this kind of object, use the formula V = l x w x h, where V is the volume, l is the length, w is the width and h is the height. For example, here's how you would find the volume of a rectangular prism with a length of four feet, a width of two feet and a height of ten feet:

V = l x w x h

V = 4 x 2 x 10

V = 80 cubic feet

Sometimes, you'll just be given the area of the rectangular prism's base and its height. Since a rectangle's length times its width equals its area, you can substitute this value into the volume formula, like this: V = b x h. In this case, b is the area of the base and h is the height. To find the volume of a rectangular prism with a height of five meters and a base that's ten square meters, you'd use the following calculations:

V = 50 cubic meters

A cylinder has a circular base, and its sides are perpendicular to this base. A can of soup is an example of a cylinder. To find the volume of a cylindrical object when you know the area of its base, you can use the formula V = b x h.

If you aren't given the area of the cylinder's base, you can find it using the formula A = (pi) r^2. A represents the area, pi equals 3.14 and r^2 is the radius of the base squared. Substitute this formula into the volume formula, and you get V = (pi) r^2 (h). For a cylinder that's three inches tall and has a base with a radius of five inches, here's how you calculate volume:

V = (pi) r^2 (h)

V = (3.14) (5^2) (3)

V = (3.14) (25) (3)

V = 235.5 cubic inches

The formula for the volume of a sphere is V = 4/3 (pi) r^3. Here's an example featuring a sphere with a radius of two centimeters:

V = 4/3 (pi) r^3

V = 4/3 (3.14) (2^3)

V = 4/3 (3.14) (8)

V = 33.493 cubic centimeters

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Home / United States / Math Classes / Formulas / Volume Formulas

Volume Formulas

The volume formulas in mathematics are used to determine the total space occupied by any three-dimensional (3-D) object. Tennis balls, dice, and even our favorite ice cream cones are some of the examples of 3D objects. Now, let's go over the volume formulas for a few 3-D shapes in depth. ...Read More Read Less

What is the Formula for Volume ?

The formula for volume is used to figure out how much space an object can hold or contain. The volume of any three-dimensional shape is measured in ‘ units\(^3\) ’ or cubic units. We have different 3-D objects in math such as cubes, cuboids, spheres, hemispheres, cones, cylinders, prisms, and pyramids. 

So, before determining the volume of any three dimensional object, it is better to have a proper idea of its formula. The volume formula for each shape is clearly shown in the table below.

Rapid Recall

Solved Examples

Ron bought a brand-new toy ball with a radius of 4 inches for his brother Chris. Find the volume of this ball.

Given, the radius of the football, \(r=4\) inches .

As we know, the shape of a ball is a sphere. 

So to find the volume of the ball, let’s use the formula to determine the volume of a sphere.

\(V=\frac{4}{3}\mathrm{\Pi}r^3\)

\(=\frac{4}{3}\times3.14\times{(4)}^3\)

\(=\frac{4}{3}\times3.14\times64\)

\(=\frac{4}{3}\times200.96\)

\(=\frac{803.84}{3}\)

\(=\) 267.94 inch\(^3\)

Therefore, the volume of Chris’s toy ball is 267.94 cubic inches.

Example 2: If the volume of a prism is 36 meter\(^3\) and the base area is 4 meter\(^2\), then, what would be the height of the prism?

Solution: As stated, the volume of a prism V = 36 meter\(^3\) and the base area is 4 meter\(^2\).

From the given data we can find the height of the prism using the formula:

V = area of base x height

36 = 4 x height

\(\frac{36}{4}\) = height

Hence, the height of the prism is 9 meters.

Example 3: Camila wants to present her mother with a nice book. She found an empty cardboard box in the store room and thoughtfully decorated it to use as a gift box for the book. But Camila is unsure whether the book which has the volume of 180 units \(^3\) will fit in the box or not. Can Camila fit the book in the box that has the dimensions of \(10\times6\times5\)?

Solution: It is stated that the dimensions of the box are \(10\times6\times5\).

From this we can say that the box is in the shape of a cuboid.

So, the dimensions can be written as l = 10, b = 6, and h = 5.

We can find the quantity the box can contain by calculating its volume.

So, the formula for the volume of a cuboid is:

= \(10\times6\times5\)

= \(60\times5\)

Therefore, the volume of the box is 300 units\(^3\).

Since, the volume of the book is 180 units\(^3\), Camila can use the box as a gift box for the book.

What is the relationship between a prism and a pyramid?

The relation between a prism and a pyramid is when a prism and pyramid have the same base and height, the volume of a pyramid is equal to 1/3 of the volume of the prism.

What is the difference between a sphere and a hemisphere?

A sphere is a circular shaped ball that has a diameter or radius. However, a hemisphere is half of a sphere. Hence, the volume of a hemisphere will be half of the volume of the sphere.

What are the real life examples of pyramids?

The pyramids of Egypt, tents, cell phone towers, a piece of a watermelon and so on are some of the examples of pyramids.

What is the difference between prisms and pyramids?

Both prisms and pyramids are three dimensional objects that have flat-faces and bases. However, a pyramid has only one polygonal base, but a prism has two identical bases.

What is the relationship between cones and cylinders?

The volume of a cone equals one-third the volume of a cylinder that has the same radius and height of the cone.

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Volume of a Sphere, Cylinder, and Cone

5 min read • december 13, 2021

Formulas and Practice to Solve for the Volume a Sphere, Cylinder, and Cone

Several similarities and differences exist between these three-dimensional figures , including having curvy (rather than flat) sides! However, each has unique features that differentiate it from another 3D object. Let’s get started!

Want the Formulas Now?

The table below displayed the volume equations for a sphere, cylinder, and cone. The information following this chart provides more in-depth information that will help you identify and distinguish one 3D object from another and practice using these formulas with a few equations.

Features of a Sphere

A sphere is a 3D object where the distance from the center point is the same . In other words, the sphere’s radius (distance from the center to the sphere’s surface) is the same in any direction in the 3D space. A sphere is entirely symmetrical, rounded, and lacks flat sides and edges.

Image courtesy of Wikipedia

Final Equation

V = 4/3𝞹r^3 , where V is the final volume and r3 represents the distance from the sphere's center to its surface, with the superscript confirming the sphere's 3D features. Understanding the concept behind the meaning of the 4/3 and 𝞹 would require the knowledge of higher-level mathematics. Regardless, it essentially encompasses a cylindrical element that contributes to the final volume of a sphere.

Features of a Cylinder

A cylinder is a prism with two circular bases and parallel sides that connect these two bases. These parallel sides that join the two flat circular ends form a tube-like shape . This object has two faces (the two circular bases) and two rims (the parallel sides that meet the rims of the two bases). Its overall shape is similar to that of a tin can.

V = 𝞹r^2h , where V is the final volume of the cylinder. Notice how rather than r^3 in the spherical equation, the cylindrical equation uses r^2 ? Since h of this equation compensates for the cylinder's height, 𝞹r2 represents the area of the circle (2D), which is why the subscript for r is 2. The height of the cylinder multiplied by the area of the flat circle produces the final volume.

Features of a Cone

A cone includes a circular base and an apex , otherwise known as the highest point of this object. The apex sits above the base's center, and there is a surface area extending from the apex to the base's border. This 3D object has one face, the base, and one edge, the rim of the circular base.

V = ⅓𝞹r^2h , where V is the final volume of 𝞹r^2 , the base of the cone, and h is the height of the cone.

Explaining 𝞹r2 

Did you notice a relationship between the cylinder and cone’s features? If you noticed that one part of the equation is 𝞹r^2 , great work! This is the same formula as that of a 2D circle's area, and multiplying this with the height produces a final volume. This is the same formula as that of a 2D circle's area, and multiplying this with the height produces a final volume.

Similarly, the volume of a rectangular prism is lwh , where lw forms the prism's base as length and width, and multiplying height h to this provides the volume.

GIF courtesy of GIPHY

Solving Problems with Spheres, Cylinders, and Cones 

If you're taking the SAT, you will likely encounter a couple of problems requiring you to solve for a missing value with limited information about that specific object. Using the equations provided above, try solving these three problems! You can find the answers and explanations at the bottom of this page. 

Problem 1 - Easy

The distance between the center and any point on the boundary of the sphere is 3 centimeters. Calculate the volume of the sphere.

Problem 2 - Medium

Juan recently purchased canned corn from the local grocery store. In preparation for his upcoming test, he decided he wanted to try and find the volume of the can. He found that the height of the can was 4.5 inches tall and 3 inches wide. What is the volume of the can in terms of pi? 

(Hint: What is the shape of a can?)

Problem 3 - Difficult

Matilda passes through construction work on the highway and wants to know the diameter of one of the traffic cones. What is the diameter of the cone if the volume is 180 in3 and the cone is 24 inches tall?

“The distance between the center and any point on the boundary of the sphere is 3 centimeters. Calculate the volume of the sphere.”

Since we know that the sphere's radius is 3 centimeters, we can plug that into our volume equation for a sphere to find the volume!

“Juan recently purchased canned corn from the local grocery store. In preparation for his upcoming test, he decided he wanted to try and find the volume of the can. He found that the height of the can was 4.5 inches tall and 3 inches wide. What is the volume of the can in terms of pi?”

We know that the height of the can is 4.5 inches, and the diameter is 3 inches. With this in mind, let’s first calculate the area of the base using the diameter, then plug that into the equation for the cylinder. Recall that A = 𝝅r^2 , and the diameter is equal to 2r . To find r , let’s divide the diameter by two: r = 3in/2 = 1.5in . Now that we have all the values we need, let’s plug it into the equation for a cylinder!

“Matilda passes through construction work on the highway and wants to know the diameter of one of the traffic cones. What is the diameter of the cone if the volume is 180in3 and the cone is 24 inches tall?”

Let’s start by plugging in the values and isolating r !

Multiply both sides by 3: 

Divide both sides by 4.92𝞹:

Find the square root of both sides:

Notice that we’re not done yet because the problem wants us to find the diameter ! Recall that r is from the center to one end of the boundary. We can use 2r to find the entire length across the base.

What to Take Away

Spheres, cylinders, and cones are incredibly important for not only solving SAT problems, but also for applicable problems in real life! When calculating the volume of a circular 3D object, the width and length use A = 𝞹r2 , which is the area of a 2D circle. Finding the relationship between 2D and 3D figures is always useful when approaching problems. Good luck on your studies! 👌

Derivation of Formula for Volume of the Sphere by Integration

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Volume – Definition, Formula, Examples

Volume is a cornerstone concept in mathematics and science that is vital for children to comprehend. It enables kids to visualize and measure the space filled by three-dimensional objects and liquids. In this comprehensive article, we will delve into the intriguing world of volume, explore various 3D shapes, and provide hands-on examples and practice problems for mastering volume-related concepts. Welcome to the exciting journey of discovering volume with Brighterly !

Volume Definition

The volume of an object or substance refers to the total amount of space it occupies in all three dimensions. It is a crucial concept that enables us to quantify how much space is taken up by objects, structures, or liquids. Volume is typically measured in units like cubic meters (m³), cubic centimeters (cm³), or liters (L). Developing a solid understanding of volume empowers children to grasp the intricate relationships between shapes and sizes, paving the way for solving real-life challenges that involve containers, liquids, architecture, and even the natural world. So, let’s embark on a fascinating journey to uncover the secrets of volume, build the foundation for further mathematical exploration, and fuel the imagination of young, bright minds!

Volume of 3-Dimensional Shapes:

When studying volume, it is essential to examine the different types of 3D shapes and how their volumes are calculated. Here are some common 3D shapes and their corresponding volume formulas:

Volume of a Cube

A cube is a 3D shape with six equal square faces. The formula for finding the volume of a cube is:

Volume = Side^3

Where “Side” represents the length of any one of the cube’s sides.

Volume of a Cylinder

A cylinder has two circular bases and a curved surface connecting them. The formula for calculating the volume of a cylinder is:

Volume = π × Radius^2 × Height

Where “Radius” is the distance from the center of the circular base to its edge, and “Height” is the distance between the two bases.

Volume of a Pyramid

A pyramid is a polyhedron with a polygonal base and triangular sides that meet at a common point, known as the apex. The formula for finding the volume of a pyramid is:

Volume = (1/3) × Base Area × Height

Where “Base Area” is the area of the polygonal base, and “Height” is the perpendicular distance from the apex to the base.

Volume of a Cone

A cone has a circular base and a curved surface that narrows to a point, called the vertex. The formula for calculating the volume of a cone is:

Volume = (1/3) × π × Radius^2 × Height

Where “Radius” is the distance from the center of the circular base to its edge, and “Height” is the distance between the vertex and the base.

Volume of a Sphere

A sphere is a perfectly round 3D shape, like a ball. The formula for finding the volume of a sphere is:

Volume = (4/3) × π × Radius^3

Where “Radius” is the distance from the center of the sphere to its surface.

Volume Worksheets Grade 7 PDF

Volume Worksheets 7th Grade PDF

Volume Worksheets 7th Grade

At Brighterly, we provide the best math worksheets for kids, where they can practice volume problems and apply their knowledge in fun and engaging ways.

Volume of Liquid

The volume of a liquid can be found by measuring the space it occupies within a container. It is typically measured in units like liters, milliliters, or gallons. Understanding liquid volume can help children solve problems involving pouring, mixing, or measuring liquids in everyday life.

Volume Formulas

Here is a summary of the volume formulas for the 3D shapes discussed above:

• Cube: Volume = Side^3
• Cylinder: Volume = π × Radius^2 × Height
• Pyramid: Volume = (1/3) × Base Area × Height
• Cone: Volume = (1/3) × π × Radius^2 × Height
• Sphere: Volume = (4/3) × π × Radius^3

How To Calculate the Volume?

• To calculate the volume of a 3D shape or liquid, follow these steps:
• Identify the shape or container.
• Determine the relevant dimensions (e.g., side length, radius, height, or base area) for the shape or container.
• Choose the appropriate volume formula for the shape.
• Plug the dimensions into the formula and perform the necessary calculations.
• Express the result in the appropriate unit of measurement (e.g., cubic meters, liters, or gallons).

Volume Related Facts

• Here are some interesting facts related to volume:
• The volume of any prism or cylinder can be found by multiplying the base area by the height.
• The volume of a cone is one-third the volume of a cylinder with the same base and height.
• The volume of a pyramid is one-third the volume of a prism with the same base and height.
• The volume of a sphere is two-thirds the volume of a cylinder that circumscribes it (i.e., completely encloses it).

Solved Examples on Volume

Example 1: cube.

A cube has a side length of 4 cm. What is its volume?

Volume = Side^3 Volume = 4^3 Volume = 64 cubic centimeters

Example 2: Cylinder

A cylinder has a radius of 3 cm and a height of 5 cm. What is its volume?

Volume = π × Radius^2 × Height Volume = π × (3^2) × 5 Volume ≈ 141.37 cubic centimeters

Volume Of Prisms Worksheet 7th Grade

Surface Area And Volume Worksheets With Answers Grade 7

Practice Questions

• Find the volume of a pyramid with a base area of 50 square centimeters and a height of 8 centimeters.
• Calculate the volume of a cone with a radius of 2 cm and a height of 6 cm.
• Determine the volume of a sphere with a radius of 5 cm.
• If a rectangular container measures 10 cm in length, 5 cm in width, and 8 cm in height, how much liquid can it hold?

Grasping the concept of volume is a critical step for children in their mathematical journey, as it unlocks the door to comprehending and solving real-world problems related to 3D shapes and liquids. Through Brighterly’s engaging and comprehensive approach, children can immerse themselves in the world of volume, fostering a deeper understanding of the diverse volume formulas and honing their problem-solving skills.

As they delve into the captivating realm of volume, young learners will not only develop the essential skills needed to tackle volume-related challenges with confidence but also ignite their curiosity to explore further mathematical concepts. With Brighterly, children will experience a dynamic and enjoyable learning process that nurtures their innate potential and empowers them to thrive in the ever-evolving world of mathematics. So, let’s embrace the adventure of discovery and growth, as we illuminate the path to a brighter future, one mathematical concept at a time!

Frequently Asked Questions on Volume

What is the difference between volume and capacity.

Volume refers to the amount of space occupied by a 3D object or substance, while capacity refers to the maximum amount of substance (usually liquid) that a container can hold. Both terms are related, but they are not interchangeable.

How do I convert volume units?

To convert between different volume units, use a conversion factor. For example, to convert from cubic meters to liters, multiply the volume in cubic meters by 1,000. To convert from liters to gallons, multiply the volume in liters by 0.26417.

Are volume and surface area related?

Volume and surface area are related in the sense that they both describe properties of 3D shapes. However, volume measures the space occupied by the shape, while surface area measures the total area of its external surfaces.

• Wikipedia – Volume
• National Council of Educational Research and Training (NCERT) – Class 9 Mathematics Textbook: Chapter 13 – Surface Areas and Volumes
• National Aeronautics and Space Administration (NASA) – Shape and Space: Geometry and Measurement
• Wolfram MathWorld – Volume

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Module 4: Equations and Inequalities

Using formulas to solve problems, learning outcomes.

• Set up a linear equation involving distance, rate, and time.
• Find the dimensions of a rectangle given the area.
• Find the dimensions of a box given information about its side lengths.

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region, [latex]A=LW[/latex]; the perimeter of a rectangle, [latex]P=2L+2W[/latex]; and the volume of a rectangular solid, [latex]V=LWH[/latex]. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

Example: Solving an Application Using a Formula

It takes Andrew 30 minutes to drive to work in the morning. He drives home using the same route, but it takes 10 minutes longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?

This is a distance problem, so we can use the formula [latex]d=rt[/latex], where distance equals rate multiplied by time. Note that when rate is given in mi/h, time must be expressed in hours. Consistent units of measurement are key to obtaining a correct solution.

First, we identify the known and unknown quantities. Andrew’s morning drive to work takes 30 min, or [latex]\frac{1}{2}[/latex] h at rate [latex]r[/latex]. His drive home takes 40 min, or [latex]\frac{2}{3}[/latex] h, and his speed averages 10 mi/h less than the morning drive. Both trips cover distance [latex]d[/latex]. A table, such as the one below, is often helpful for keeping track of information in these types of problems.

Write two equations, one for each trip.

As both equations equal the same distance, we set them equal to each other and solve for r .

We have solved for the rate of speed to work, 40 mph. Substituting 40 into the rate on the return trip yields 30 mi/h. Now we can answer the question. Substitute the rate back into either equation and solve for d.

The distance between home and work is 20 mi.

Analysis of the Solution

Note that we could have cleared the fractions in the equation by multiplying both sides of the equation by the LCD to solve for [latex]r[/latex].

On Saturday morning, it took Jennifer 3.6 hours to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 hours to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday?

45 [latex]\frac{\text{mi}}{\text{h}}[/latex]

Example: Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is [latex]54[/latex] ft. The length is [latex]3[/latex] ft. greater than the width. What are the dimensions of the patio?

The perimeter formula is standard: [latex]P=2L+2W[/latex]. We have two unknown quantities, length and width. However, we can write the length in terms of the width as [latex]L=W+3[/latex]. Substitute the perimeter value and the expression for length into the formula. It is often helpful to make a sketch and label the sides as shown below.

Now we can solve for the width and then calculate the length.

The dimensions are [latex]L=15[/latex] ft and [latex]W=12[/latex] ft.

Find the dimensions of a rectangle given that the perimeter is [latex]110[/latex] cm. and the length is 1 cm. more than twice the width.

L = 37 cm, W = 18 cm

Example: Solving an Area Problem

The perimeter of a tablet of graph paper is 48 in. The length is [latex]6[/latex] in. more than the width. Find the area of the graph paper.

The standard formula for area is [latex]A=LW[/latex]; however, we will solve the problem using the perimeter formula. The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns. As both perimeter and area use length and width as dimensions, they are often used together to solve a problem such as this one.

We know that the length is 6 in. more than the width, so we can write length as [latex]L=W+6[/latex]. Substitute the value of the perimeter and the expression for length into the perimeter formula and find the length.

Now, we find the area given the dimensions of [latex]L=15[/latex] in. and [latex]W=9[/latex] in.

The area is [latex]135[/latex] in 2 .

A game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft 2 of new carpeting should be ordered?

Example: Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is [latex]8[/latex] inches, and the volume is 1,600 in. 3 .

The formula for the volume of a box is given as [latex]V=LWH[/latex], the product of length, width, and height. We are given that [latex]L=2W[/latex], and [latex]H=8[/latex]. The volume is [latex]1,600[/latex] cubic inches.

The dimensions are [latex]L=20[/latex] in., [latex]W=10[/latex] in., and [latex]H=8[/latex] in.

Note that the square root of [latex]{W}^{2}[/latex] would result in a positive and a negative value. However, because we are describing width, we can use only the positive result.

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5th Grade Volume Worksheets

Welcome to our 5th Grade Volume Worksheets page.

Here you will find our collection of worksheets to introduce and help you learn about volume.

These worksheets will help you to understand and practice how to find the volume of rectangular prisms and other simple shapes.

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Volume of Rectangular Prisms

On this webpage you will find our range of worksheets to help you work out the volume of simple 3d shapes such as rectangular prisms.

They are also very useful for introducing the concept of volume being the number of cubes that fill up a space.

These sheets are graded from easiest to hardest, and each sheet comes complete with answers.

Using these sheets will help your child to:

• learn how to find the volume of simple 3d shapes by counting cubes;
• learn how to find the volume of rectangular prisms by multiplying length x width x height
• practice using their knowledge to solve basic volume problems.

What is Volume?

• Volume is the amount of space that is inside a 3 dimensional shape.
• Because it is an amount of space, it has to be measured in cubes.
• If the shape is measured in cm, then the volume would be measured in cubic cm or cm 3
• If the shape is measured in inches, then the volume would be measured in cubic inches or in 3

Volume of a Rectangular Prism

• The volume of a rectangular prism is the number of cubes it is made from.
• To find the number of cubes, we need to multiply the length by the width by the height.
• So Volume = length x width x height or l x w x h.
• We could also multiply the area of the base (which is the length x width) by the height.
• So Volume = l x w x h or b x h (where b is the area of the base)

In the example above, the length is 3, the width is 6 and the height is 2.

So the volume is 3 x 6 x 2 = 36cm 3 or 36 cubic cm.

This tells us that there are 36 cm cubes that make up the shape.

We have split our worksheets up into different sections, to make it easier for you to select the right sheets for your needs.

• Section 1 - Find the Volume by Counting Cubes
• Section 2 - Finding the Volume by multiplication
• Section 3 - Match the Volume (multiplication)
• Section 3 - Volume Problem Solving Riddles

5th Grade Volume Worksheets - Counting Cubes

• Volume - Count the Cubes Sheet 1
• PDF version
• Volume - Count the Cubes Sheet 2

5th Grade Volume Worksheets - Find the Volume by Multiplication

The first sheet is supported, the other two sheets are more independent.

You can choose between the standard or metric versions of sheets 2 and 3 (the measurements are the same)

• Find the Volume Sheet 1 (supported)
• Find the Volume Sheet 2 (standard)
• Find the Volume Sheet 2 (metric)
• Find the Volume Sheet 3 (standard)
• Find the Volume Sheet 3 (metric)

5th Grade Volume Worksheets - Match the Volume

• Match the Volume Sheet 1
• Match the Volume Sheet 2

5th Grade Volume Worksheets - Volume Riddles

• Volume Riddles Sheet 5A
• Volume Riddles Sheet 5B

Volume of Rectangular Prisms Walkthrough Video

This short video walkthrough shows several problems from our Find the Volume Sheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, check out the video below!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Volume of a Cube/Cuboid/Box Calculators

Each of the pages below includes a handy calculator to help you find the volume of cubes, cuboids and boxes.

• Volume of a Cube Calculator

• Volume of a Box Calculator

Converting Measures Worksheets

Here is our selection of converting units of measure for 3rd to 5th graders.

These sheets involve converting between customary units of measure and also metric units.

• Converting Customary Units Worksheets
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5th Grade Geometry Worksheets

Here is our selection of 5th grade Geometry worksheets about angles.

The focus on these sheets is angles on a straight line, angles around a point and angles in a triangle.

• 5th Grade Geometry Missing Angles

Area Worksheets

Here is our selection of free printable area worksheets for 3rd and 4th grade.

The sheets are all graded in order from easiest to hardest.

• work out the areas of a range of rectangles;
• find the area of rectilinear shapes.
• Area Worksheets - Rectangles and Rectilinear Shapes
• Area of Right Triangles
• Area of Quadrilaterals Worksheets
• Perimeter Worksheets

Here is our selection of free printable perimeter worksheets for 3rd and 4th grade.

• work out the perimeter of a range of rectangles;
• find the perimeter of rectilinear shapes.

All the math practice worksheets in this section support Elementary Math Benchmarks.

How to Print or Save these sheets

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Volume Word Problems Worksheets

How to Find the Volume of Basic Shapes - In geometry, we need to figure out the volume, surface area, and perimeter of the shapes. There are a number of shapes in geometry; each shape has a specific formula for perimeter, surface area, and volume. We should follow those formulae to find out the specific measurement of the shape. Here we are going to discuss the formulae of finding the volume of some of the geometric shapes. Volume of the Sphere - A sphere is a three-dimensional shape. To find out the surface area or volume of the sphere, we need to know the radius of the sphere. The radius of the sphere is the distance from the center to the edge of the sphere. The radius remains the same no matter from which point on a sphere is considered. Once we know the radius, we use the following formula to find out the volume of the sphere. Volume = 4/3 πr 2 . Volume of the Cone - A cone is defined as the pyramid that features a circular base with sloping sides, all meeting at the central point. For calculating the volume of the cone, we need to know the radius of length and base of the side. Volume = 1/3 π 2 h. Volume of the Cylinder - A cylinder is a shape that features a circular base and parallel sides. For calculating the volume of the cylinder, we need to know the height and radius of the cylinder. Volume = πr 2 h.

Basic Lesson

Demonstrates how to outline Volume Word Problems. Example: Find the volume of cube with 7 cm sides. Volume of cube = (side) 3

Intermediate Lesson

Uses slightly larger sentences and numbers than the basic lesson. Example: Find the volume of sphere of radius of 21 inches. Volume of sphere = 4/3 × π × r 3

Independent Practice 1

Contains a series of 20 volume Word Problems. The answers can be found below. Example: A large cylindrical can is to be designed from a rectangular piece of aluminum that is 25 inches long and 10 inches high by rolling the metal horizontally. Determine the volume of the cylinder.

Independent Practice 2

Features 20 word problems. Example: The dimensions of a chamber are 11 ft by 9 ft by 9 ft. The chamber is thought to have been used for storing ammunition had dimensions of 1 ft by 1 ft by 3 ft. What is the maximum number of ammunition boxes of that size that could be put in the underground chamber?

Homework Worksheet

10 word problems for students to work on at home. An example problem is provided and explained. Example: Suppose a swimming pool in the shape of a hemisphere is 28m wide. How much water can the pool hold? Round your answers to one decimal place.

10 volume based Word Problems. A math scoring matrix is included. Example: What is the volume of a regular cylinder whose base has radius of 16 cm and has height of 35 cm?

Homework and Quiz Answer Key

Answers for the homework and quiz.

Answers for the lesson and practice sheets.

Find Their Volume....

A mathematician, a physicist, and an engineer are all given identical rubber balls and told to find the volume. They are given anything they want to measure it, and have all the time they need. The mathematician pulls out a measuring tape and records the circumference. He then divides by two times pi to get the radius, cubes that, multiplies by pi again, and then multiplies by four-thirds and thereby calculates the volume. The physicist gets a bucket of water, places 1.00000 gallons of water in the bucket, drops in the ball, and measures the displacement to six significant figures. And the engineer? He writes down the serial number of the ball, and looks it up.