Find prism and cylinder volume from base area and height

Ontario Grade 7 Mathematics

Spatial Sense

Finding the volume of prisms and cylinders

Imagine stacking identical sheets of paper. One sheet covers a certain area, and the stack has a certain height. The space inside the stack depends on both: the area of one sheet and how many sheets fit from bottom to top. A prism or cylinder works in the same way. In this lesson, the base area is given, so you do not need to find the area of a particular shape. You will multiply the base area by the solid’s height.

What you will learn

Explain how base area and height determine the volume of a prism or cylinder.
Find volume when the base area and height are given.
Label volume with cubic units and check that the answer makes sense.

From one layer to a whole solid

A solid is a three-dimensional shape. Its volume is the amount of space it takes up. We measure volume in cubic units, such as cubic centimetres (

\text{cm}^3

).

The base is the flat face used as the bottom of the solid. The base area tells how much space that face covers. Height is the straight distance from the base to the matching top face. For a cylinder, height is the distance between its two circular bases.

Think of a stack of identical square tiles. If each layer covers

12\text{ cm}^2

and there are 4 layers, the stack has a volume of

48\text{ cm}^3

. The same idea applies when the base is a different shape, including a circle. Prisms and cylinders have the same-sized, parallel bases all the way through their height.

The rule is to multiply the area of one base by the height. The given base area can be used directly. Check that its units are square units and that height uses the matching length unit. The answer then uses cubic units.

V=Bh

Base area describes one end or layer of the solid.
Height measures the distance between the two matching bases.
Multiply base area by height; use cubic units for volume.

Use the rule for a prism or cylinder

A prism is a solid with two matching, parallel ends and flat side faces. Its ends can have different shapes. A cylinder has two matching, parallel circular bases and a curved side. Both solids have the same volume rule because each has the same base area through its height.

In the rule,

V

means volume,

B

means the area of one base, and

h

means height. For example, if

B

is in square centimetres and

h

is in centimetres, then

V

is in cubic centimetres.

A labelled drawing can help you identify the base and height. The drawing below represents a prism with a base area of

18\text{ cm}^2

and a height of

4\text{ cm}

. Its volume is the area of that base repeated through the height. The drawing is a model of the stated measurements, not a scale drawing.

Before calculating, ask: What is the base area? What is the height? Do their units match? After calculating, write the unit as cubic units. Do not use only the base area as the volume; the height matters too.

V=Bh

The same rule works for prisms and cylinders.
Use the stated base area, even when the base is circular.
Square units multiplied by length units give cubic units.

Try a sensible estimate and check

Volume should grow when either the base area or height grows. If the base area doubles while the height stays the same, the volume doubles. If the height doubles while the base area stays the same, the volume also doubles.

A quick estimate can help catch an arithmetic error. If a base area is about

20\text{ cm}^2

and the height is about

5\text{ cm}

, the volume should be about

100\text{ cm}^3

. A result of

10\text{ cm}^3

would be much too small for those measurements.

When you practise, show the multiplication and include units in the final answer. For instance, find a prism’s volume if its base area is

16\text{ m}^2

and its height is

3\text{ m}

. Then check: does the answer use cubic metres? The worked examples show the full process for different given base areas.

A larger base area or height leads to a larger volume, when the other measurement stays fixed.
Estimate to check whether the answer is reasonable.
Include cubic units in every final volume.

Worked example

A prism with a rectangular base

A prism has a base area of

24\text{ cm}^2

and a height of

5\text{ cm}

. Find its volume.

Prism with stated base area and height

The base area is 24 cm² and the perpendicular distance between the matching bases is 5 cm. The drawing is schematic.

Identify the measurements
The base area is $24\text{ cm}^2$ , and the height is $5\text{ cm}$ . The units are compatible: square centimetres times centimetres will give cubic centimetres.
$B=24\text{ cm}^2,\ h=5\text{ cm}$
Multiply base area by height
Use the volume rule. Multiplying the area of one base by the height counts the space through the full prism.
$V=24\times5=120\text{ cm}^3$

Answer: The prism’s volume is

120\text{ cm}^3

.

Check: The height is 5 cm, so the volume is five layers of 24 cm²:

5\times24=120\text{ cm}^3

.

Worked example

A prism with a different-shaped base

A prism has a base area of

15\text{ cm}^2

and a height of

8\text{ cm}

. Find its volume.

Prism with stated base area and height

The base area is 15 cm² and the perpendicular distance between the matching bases is 8 cm. The drawing is schematic.

Use the given base area
The base’s shape does not change the volume rule. Since its area is already given, use $15\text{ cm}^2$ directly.
$B=15\text{ cm}^2,\ h=8\text{ cm}$
Calculate and label
Multiply the base area by the height. The product uses cubic centimetres because the measurements are in square centimetres and centimetres.
$V=15\times8=120\text{ cm}^3$

Answer: The prism’s volume is

120\text{ cm}^3

.

Check: The result is greater than the base area because the prism extends through a height of 8 cm. Its unit is cubic centimetres.

Worked example

A cylinder with base area supplied

A cylinder has a base area of

50.24\text{ cm}^2

and a height of

6\text{ cm}

. Find its volume.

Cylinder with stated base area and height

The base area is 50.24 cm² and the perpendicular distance between the circular bases is 6 cm. The drawing is schematic.

Read the base area and height
The base is circular, but its area is already supplied. There is no need to find the circle’s area. Use the cylinder’s height of 6 cm.
$B=50.24\text{ cm}^2,\ h=6\text{ cm}$
Multiply to find volume
A cylinder has the same base area through its height, so multiply the given area by the height. The product is in cubic centimetres.
$V=50.24\times6=301.44\text{ cm}^3$

Answer: The cylinder’s volume is

301.44\text{ cm}^3

.

Check: Since the height is 6 cm, the answer is six times the base area:

50.24\times6=301.44

.

Common mistakes and how to avoid them

Writing the base area as the volume.

Correction: Base area describes only one base. Multiply it by the height to find the volume.

Writing square units in the final answer.

Correction: Volume uses cubic units. For example, use

\text{cm}^3

, not

\text{cm}^2

.

Trying to find a cylinder’s circular base area even though it is given.

Correction: Use the supplied base area directly in

V=Bh

.

Using a height unit that does not match the base-area unit.

Correction: Check that the length unit in the height matches the length unit used for the base area before multiplying.

Lesson summary

Volume measures the space inside a solid.
For a prism or cylinder, multiply base area by height.
Use cubic units, such as $\text{cm}^3$ , for the answer.
When base area is given, use it directly.

Check your understanding

Question 1

A prism has base area

9\text{ m}^2

and height

7\text{ m}

. What is its volume?

$16\text{ m}^3$
$63\text{ m}^3$
$63\text{ m}^2$
$7\text{ m}^3$

Show answer and explanation

63\text{ m}^3

Multiply the base area by the height:

9\times7=63

. The volume unit is cubic metres.

Question 2

A cylinder has base area

32\text{ cm}^2

and height

4\text{ cm}

. Which calculation finds its volume?

$32+4$
$32\times4$
$32\div4$
$32-4$

Show answer and explanation

32\times4

The volume rule is base area multiplied by height. The volume is

128\text{ cm}^3

.

Question 3

A prism’s base area is

11\text{ cm}^2

and its height is

3\text{ cm}

. Which unit belongs with its volume?

$\text{cm}$
$\text{cm}^2$
$\text{cm}^3$
No unit

Show answer and explanation

\text{cm}^3

Square centimetres multiplied by centimetres give cubic centimetres. The volume is

33\text{ cm}^3

.

Key terms

Volume: The amount of space inside a three-dimensional solid.
Base: A flat end or face used as the bottom of a prism or cylinder.
Base area: The amount of surface covered by one base.
Height: The straight distance between the matching bases.
Prism: A solid with two matching, parallel ends and flat side faces.
Cylinder: A solid with two matching, parallel circular bases and a curved side.

Continue through Grade 7 Math

View the complete Grade 7 Math Ontario Grade 7 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation E2.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

E2.7 · Find prism and cylinder volume from base area and height

Ontario Grade 7 Mathematics

Finding the volume of prisms and cylinders

What you will learn

From one layer to a whole solid

Use the rule for a prism or cylinder

Try a sensible estimate and check

A prism with a rectangular base

A prism with a different-shaped base

A cylinder with base area supplied

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review