Contributed by:

The volume of a solid is the measure of how much space an object takes up. It is measured by the number of unit cubes it takes to fill up the solid.

1.
4.3 Volumes of Solids

Learning Objective(s)

1 Identify geometric solids.

2 Find the volume of geometric solids.

3 Find the volume of a composite geometric solid.

Living in a two-dimensional world would be pretty boring. Thankfully, all of the physical

objects that you see and use every day—computers, phones, cars, shoes—exist in three

dimensions. They all have length, width, and height. (Even very thin objects like a piece

of paper are three-dimensional. The thickness of a piece of paper may be a fraction of a

millimeter, but it does exist.)

In the world of geometry, it is common to see three-dimensional figures. In mathematics,

a flat side of a three-dimensional figure is called a face. Polyhedrons are shapes that

have four or more faces, each one being a polygon. These include cubes, prisms, and

pyramids. Sometimes you may even see single figures that are composites of two of

these figures. Let’s take a look at some common polyhedrons.

Identifying Solids Objective 1

The first set of solids contains rectangular bases. Have a look at the table below, which

shows each figure in both solid and transparent form.

Name Definition Solid Form Transparent Form

A six-sided

polyhedron that has

congruent squares

as faces.

A polyhedron that

has three pairs of

Rectangular prism congruent,

rectangular, parallel

faces.

A polyhedron with a

polygonal base and

Pyramid a collection of

triangular faces that

meet at a point.

4.23

Learning Objective(s)

1 Identify geometric solids.

2 Find the volume of geometric solids.

3 Find the volume of a composite geometric solid.

Living in a two-dimensional world would be pretty boring. Thankfully, all of the physical

objects that you see and use every day—computers, phones, cars, shoes—exist in three

dimensions. They all have length, width, and height. (Even very thin objects like a piece

of paper are three-dimensional. The thickness of a piece of paper may be a fraction of a

millimeter, but it does exist.)

In the world of geometry, it is common to see three-dimensional figures. In mathematics,

a flat side of a three-dimensional figure is called a face. Polyhedrons are shapes that

have four or more faces, each one being a polygon. These include cubes, prisms, and

pyramids. Sometimes you may even see single figures that are composites of two of

these figures. Let’s take a look at some common polyhedrons.

Identifying Solids Objective 1

The first set of solids contains rectangular bases. Have a look at the table below, which

shows each figure in both solid and transparent form.

Name Definition Solid Form Transparent Form

A six-sided

polyhedron that has

congruent squares

as faces.

A polyhedron that

has three pairs of

Rectangular prism congruent,

rectangular, parallel

faces.

A polyhedron with a

polygonal base and

Pyramid a collection of

triangular faces that

meet at a point.

4.23

2.
Notice the different names that are used for these figures. A cube is different than a

square, although they are sometimes confused with each other—a cube has three

dimensions, while a square only has two. Likewise, you would describe a shoebox as a

rectangular prism (not simply a rectangle), and the ancient pyramids of Egypt

as…well, as pyramids (not triangles)!

In this next set of solids, each figure has a circular base.

Name Definition Solid Form Transparent Form

A solid figure with a

pair of circular,

parallel bases and a

round, smooth face

between them.

A solid figure with a

single circular base

and a round,

smooth face that

diminishes to a

single point.

Take a moment to compare a pyramid and a cone. Notice that a pyramid has a

rectangular base and flat, triangular faces; a cone has a circular base and a smooth,

rounded body.

Finally, let’s look at a shape that is unique: a sphere.

Name Definition Solid Form Transparent Form

A solid, round figure

where every point

on the surface is the

same distance from

Sphere the center.

There are many spherical objects all around you—soccer balls, tennis balls, and

baseballs being three common items. While they may not be perfectly spherical, they are

generally referred to as spheres.

4.24

square, although they are sometimes confused with each other—a cube has three

dimensions, while a square only has two. Likewise, you would describe a shoebox as a

rectangular prism (not simply a rectangle), and the ancient pyramids of Egypt

as…well, as pyramids (not triangles)!

In this next set of solids, each figure has a circular base.

Name Definition Solid Form Transparent Form

A solid figure with a

pair of circular,

parallel bases and a

round, smooth face

between them.

A solid figure with a

single circular base

and a round,

smooth face that

diminishes to a

single point.

Take a moment to compare a pyramid and a cone. Notice that a pyramid has a

rectangular base and flat, triangular faces; a cone has a circular base and a smooth,

rounded body.

Finally, let’s look at a shape that is unique: a sphere.

Name Definition Solid Form Transparent Form

A solid, round figure

where every point

on the surface is the

same distance from

Sphere the center.

There are many spherical objects all around you—soccer balls, tennis balls, and

baseballs being three common items. While they may not be perfectly spherical, they are

generally referred to as spheres.

4.24

3.
Example

Problem A three-dimensional figure has the following properties:

• It has a rectangular base.

• It has four triangular faces.

What kind of a solid is it?

A rectangular

base indicates

that it must be a

cube, rectangular

prism, or pyramid.

Since the faces

are triangular, it

must be a

pyramid.

Answer The solid is a pyramid.

Volume Objective 2

Recall that perimeter measures one dimension (length), and area measures two

dimensions (length and width). To measure the amount of space a three-dimensional

figure takes up, you use another measurement called volume.

To visualize what “volume” measures, look back at the transparent image of the

rectangular prism mentioned earlier (or just think of an empty shoebox). Imagine

stacking identical cubes inside that box so that there are no gaps between any of the

cubes. Imagine filling up the entire box in this manner. If you counted the number of

cubes that fit inside that rectangular prism, you would have its volume.

Volume is measured in cubic units. The shoebox illustrated above may be measured in

cubic inches (usually represented as in3 or inches3), while the Great Pyramid of Egypt

would be more appropriately measured in cubic meters (m3 or meters3).

4.25

Problem A three-dimensional figure has the following properties:

• It has a rectangular base.

• It has four triangular faces.

What kind of a solid is it?

A rectangular

base indicates

that it must be a

cube, rectangular

prism, or pyramid.

Since the faces

are triangular, it

must be a

pyramid.

Answer The solid is a pyramid.

Volume Objective 2

Recall that perimeter measures one dimension (length), and area measures two

dimensions (length and width). To measure the amount of space a three-dimensional

figure takes up, you use another measurement called volume.

To visualize what “volume” measures, look back at the transparent image of the

rectangular prism mentioned earlier (or just think of an empty shoebox). Imagine

stacking identical cubes inside that box so that there are no gaps between any of the

cubes. Imagine filling up the entire box in this manner. If you counted the number of

cubes that fit inside that rectangular prism, you would have its volume.

Volume is measured in cubic units. The shoebox illustrated above may be measured in

cubic inches (usually represented as in3 or inches3), while the Great Pyramid of Egypt

would be more appropriately measured in cubic meters (m3 or meters3).

4.25

4.
To find the volume of a geometric solid, you could create a transparent version of the

solid, create a bunch of 1x1x1 cubes, and then stack them carefully inside. However,

that would take a long time! A much easier way to find the volume is to become familiar

with some geometric formulas, and to use those instead.

Let’s go through the geometric solids once more and list the volume formula for each.

As you look through the list below, you may notice that some of the volume formulas

look similar to their area formulas. To find the volume of a rectangular prism, you find the

area of the base and then multiply that by the height.

Name Transparent Form Volume Formula

= • a • a a3

V a=

Cube

a = the length of

one side

V = l •w • h

Rectangular prism l = length

w = width

h = height

l •w • h

V=

3

Pyramid

l = length

w = width

h = height

Remember that all cubes are rectangular prisms, so the formula for finding the volume of

a cube is the area of the base of the cube times the height.

4.26

solid, create a bunch of 1x1x1 cubes, and then stack them carefully inside. However,

that would take a long time! A much easier way to find the volume is to become familiar

with some geometric formulas, and to use those instead.

Let’s go through the geometric solids once more and list the volume formula for each.

As you look through the list below, you may notice that some of the volume formulas

look similar to their area formulas. To find the volume of a rectangular prism, you find the

area of the base and then multiply that by the height.

Name Transparent Form Volume Formula

= • a • a a3

V a=

Cube

a = the length of

one side

V = l •w • h

Rectangular prism l = length

w = width

h = height

l •w • h

V=

3

Pyramid

l = length

w = width

h = height

Remember that all cubes are rectangular prisms, so the formula for finding the volume of

a cube is the area of the base of the cube times the height.

4.26

5.
Now let’s look at solids that have a circular base.

Name Transparent Form Volume Formula

V =π •r2 •h

Cylinder

r = radius

h = height

π •r2 •h

V=

3

Cone

r = radius

h = height

Here you see the number π again.

The volume of a cylinder is the area of its base, π r 2 , times its height, h.

π •r2 •h

Compare the formula for the volume of a cone ( V = ) with the formula for the

3

l •w • h

volume of a pyramid (V = ). The numerator of the cone formula is the volume

3

formula for a cylinder, and the numerator of the pyramid formula is the volume formula

for a rectangular prism. Then divide each by 3 to find the volume of the cone and the

pyramid. Looking for patterns and similarities in the formulas can help you remember

which formula refers to a given solid.

Finally, the formula for a sphere is provided below. Notice that the radius is cubed, not

4

squared and that the quantity π r 3 is multiplied by .

3

Name Wireframe Form Volume Formula

4 3

V= πr

Sphere 3

r = radius

4.27

Name Transparent Form Volume Formula

V =π •r2 •h

Cylinder

r = radius

h = height

π •r2 •h

V=

3

Cone

r = radius

h = height

Here you see the number π again.

The volume of a cylinder is the area of its base, π r 2 , times its height, h.

π •r2 •h

Compare the formula for the volume of a cone ( V = ) with the formula for the

3

l •w • h

volume of a pyramid (V = ). The numerator of the cone formula is the volume

3

formula for a cylinder, and the numerator of the pyramid formula is the volume formula

for a rectangular prism. Then divide each by 3 to find the volume of the cone and the

pyramid. Looking for patterns and similarities in the formulas can help you remember

which formula refers to a given solid.

Finally, the formula for a sphere is provided below. Notice that the radius is cubed, not

4

squared and that the quantity π r 3 is multiplied by .

3

Name Wireframe Form Volume Formula

4 3

V= πr

Sphere 3

r = radius

4.27

6.
Applying the Formulas

You know how to identify the solids, and you also know the volume formulas for these

solids. To calculate the actual volume of a given shape, all you need to do is substitute

the solid’s dimensions into the formula and calculate.

In the examples below, notice that cubic units (meters3, inches3, feet3) are used.

Example

Problem Find the volume of a cube with side lengths of 6

meters.

= • a • a a3 Identify the proper

V a=

formula to use.

a = side length

= • 6 • 6 63 Substitute a = 6 into the

V 6=

formula.

6 • 6 • 6 = 216 Calculate the volume.

Answer Volume = 216 meters3

Example

Problem Find the volume of the shape shown below.

Pyramid. Identify the shape. It has

a rectangular base and

rises to a point, so it is a

pyramid.

l • w • h Identify the proper

V= formula to use.

3

4.28

You know how to identify the solids, and you also know the volume formulas for these

solids. To calculate the actual volume of a given shape, all you need to do is substitute

the solid’s dimensions into the formula and calculate.

In the examples below, notice that cubic units (meters3, inches3, feet3) are used.

Example

Problem Find the volume of a cube with side lengths of 6

meters.

= • a • a a3 Identify the proper

V a=

formula to use.

a = side length

= • 6 • 6 63 Substitute a = 6 into the

V 6=

formula.

6 • 6 • 6 = 216 Calculate the volume.

Answer Volume = 216 meters3

Example

Problem Find the volume of the shape shown below.

Pyramid. Identify the shape. It has

a rectangular base and

rises to a point, so it is a

pyramid.

l • w • h Identify the proper

V= formula to use.

3

4.28

7.
l = length = 4 Use the image to identify

w = width = 3 the dimensions. Then

h = height = 8 substitute l = 4, w = 3,

and h = 8 into the

4 • 3 • 8 formula.

V=

3

96 Calculate the volume.

V=

3

= 32

Answer The volume of the pyramid is 32 inches3.

Example

Problem Find the volume of the shape shown below.

Use 3.14 for π , and round the answer to the nearest

hundredth.

Cylinder. Identify the shape. It has

a circular base and has

uniform thickness (or

height), so it is a cylinder.

V = π • r 2 • h Identify the proper

formula to use.

V = π • 72 • 1 Use the image to identify

the dimensions. Then

substitute r = 7 and h = 1

into the formula.

V = π • 49 • 1 Calculate the volume,

using 3.14 as an

= 49π

approximation for π .

≈ 153.86

Answer The volume is 49 π or approximately 153.86 feet3.

Self Check A

Find the volume of a rectangular prism that is 8 inches long, 3 inches wide, and 10

inches tall.

4.29

w = width = 3 the dimensions. Then

h = height = 8 substitute l = 4, w = 3,

and h = 8 into the

4 • 3 • 8 formula.

V=

3

96 Calculate the volume.

V=

3

= 32

Answer The volume of the pyramid is 32 inches3.

Example

Problem Find the volume of the shape shown below.

Use 3.14 for π , and round the answer to the nearest

hundredth.

Cylinder. Identify the shape. It has

a circular base and has

uniform thickness (or

height), so it is a cylinder.

V = π • r 2 • h Identify the proper

formula to use.

V = π • 72 • 1 Use the image to identify

the dimensions. Then

substitute r = 7 and h = 1

into the formula.

V = π • 49 • 1 Calculate the volume,

using 3.14 as an

= 49π

approximation for π .

≈ 153.86

Answer The volume is 49 π or approximately 153.86 feet3.

Self Check A

Find the volume of a rectangular prism that is 8 inches long, 3 inches wide, and 10

inches tall.

4.29

8.
Composite Solids Objective 3

Composite geometric solids are made from two or more geometric solids. You can find

the volume of these solids as well, as long as you are able to figure out the individual

solids that make up the composite shape.

Look at the image of a capsule below. Each end is a half-sphere. You can find the

volume of the solid by taking it apart. What solids can you break this shape into?

You can break it into a cylinder and two half-spheres.

Two half-spheres form a whole one, so if you know the volume formulas for a cylinder

and a sphere, you can find the volume of this capsule.

Example

Problem If the radius of the spherical ends is 6 inches, find the volume of the

solid below. Use 3.14 for π . Round your final answer to the nearest

whole number.

4.30

Composite geometric solids are made from two or more geometric solids. You can find

the volume of these solids as well, as long as you are able to figure out the individual

solids that make up the composite shape.

Look at the image of a capsule below. Each end is a half-sphere. You can find the

volume of the solid by taking it apart. What solids can you break this shape into?

You can break it into a cylinder and two half-spheres.

Two half-spheres form a whole one, so if you know the volume formulas for a cylinder

and a sphere, you can find the volume of this capsule.

Example

Problem If the radius of the spherical ends is 6 inches, find the volume of the

solid below. Use 3.14 for π . Round your final answer to the nearest

whole number.

4.30

9.
Identify the composite solids.

This capsule can be thought

of as a cylinder with a half-

sphere on each end.

Volume of a cylinder: π • r 2 • h Identify the proper formulas

4 3 to use.

Volume of a sphere: π r

3

Volume of a cylinder: π • 62 • 24 Substitute the dimensions

4 into the formulas.

Volume of a sphere: π • 63

3

The height of a cylinder

refers to the section between

the two circular bases. This

dimension is given as 24

inches, so h = 24.

The radius of the sphere is 6

inches. You can use r = 6 in

both formulas.

V = π • 36 • 24 Calculate the volume of the

cylinder and the sphere.

Volume of the cylinder: = 864 • π

≈ 2712.96

4

V= π • 216

3

Volume of the sphere: = 288 • π

≈ 904.32

Volume of capsule: Add the volumes.

2712.96 + 904.32 ≈ 3617.28

Answer The volume of the capsule is 1,152 π or approximately 3617 inches3.

4.31

This capsule can be thought

of as a cylinder with a half-

sphere on each end.

Volume of a cylinder: π • r 2 • h Identify the proper formulas

4 3 to use.

Volume of a sphere: π r

3

Volume of a cylinder: π • 62 • 24 Substitute the dimensions

4 into the formulas.

Volume of a sphere: π • 63

3

The height of a cylinder

refers to the section between

the two circular bases. This

dimension is given as 24

inches, so h = 24.

The radius of the sphere is 6

inches. You can use r = 6 in

both formulas.

V = π • 36 • 24 Calculate the volume of the

cylinder and the sphere.

Volume of the cylinder: = 864 • π

≈ 2712.96

4

V= π • 216

3

Volume of the sphere: = 288 • π

≈ 904.32

Volume of capsule: Add the volumes.

2712.96 + 904.32 ≈ 3617.28

Answer The volume of the capsule is 1,152 π or approximately 3617 inches3.

4.31

10.
Self Check B

A machine takes a solid cylinder with a height of 9 mm and a

diameter of 7 mm, and bores a hole all the way through it. The

hole that it creates has a diameter of 3 mm. Find the volume

of the solid.

Three-dimensional solids have length, width, and height. You use a measurement called

volume to figure out the amount of space that these solids take up. To find the volume of

a specific geometric solid, you can use a volume formula that is specific to that solid.

Sometimes, you will encounter composite geometric solids. These are solids that

combine two or more basic solids. To find the volume of these, identify the simpler solids

that make up the composite figure, find the volumes of those solids, and combine them

as needed.

4.3 Self Check Solutions

Self Check A

Find the volume of a rectangular prism that is 8 inches long, 3 inches wide, and 10

inches tall.

240 inches3

To find the volume of the rectangular prism, use the formula V = l • w • h , and then

substitute in the values for the length, width, and height. 8 inches • 3 inches • 10 inches

= 240 inches3.

Self Check B

A machine takes a solid cylinder with a height of 9 mm and a diameter of 7 mm, and

bores a hole all the way through it. The hole that it creates has a diameter of 3 mm. Find

the volume of the solid.

You find the volume of the entire cylinder by multiplying π • 3.52 • 9 , then subtract the

empty cylinder in the middle, which is found by multiplying π • 1.52 • 9 .

(π • 3.52 • 9) − (π • 1.52 • 9) ≈ 282.6mm3

4.32

A machine takes a solid cylinder with a height of 9 mm and a

diameter of 7 mm, and bores a hole all the way through it. The

hole that it creates has a diameter of 3 mm. Find the volume

of the solid.

Three-dimensional solids have length, width, and height. You use a measurement called

volume to figure out the amount of space that these solids take up. To find the volume of

a specific geometric solid, you can use a volume formula that is specific to that solid.

Sometimes, you will encounter composite geometric solids. These are solids that

combine two or more basic solids. To find the volume of these, identify the simpler solids

that make up the composite figure, find the volumes of those solids, and combine them

as needed.

4.3 Self Check Solutions

Self Check A

Find the volume of a rectangular prism that is 8 inches long, 3 inches wide, and 10

inches tall.

240 inches3

To find the volume of the rectangular prism, use the formula V = l • w • h , and then

substitute in the values for the length, width, and height. 8 inches • 3 inches • 10 inches

= 240 inches3.

Self Check B

A machine takes a solid cylinder with a height of 9 mm and a diameter of 7 mm, and

bores a hole all the way through it. The hole that it creates has a diameter of 3 mm. Find

the volume of the solid.

You find the volume of the entire cylinder by multiplying π • 3.52 • 9 , then subtract the

empty cylinder in the middle, which is found by multiplying π • 1.52 • 9 .

(π • 3.52 • 9) − (π • 1.52 • 9) ≈ 282.6mm3

4.32