Apply volume relationships for prisms, pyramids, cylinders, and cones

Ontario Grade 9 Mathematics

Geometry and Measurement

How shape and matching dimensions affect the space a solid holds

Imagine filling a straight-sided container with small cubes. The number of cubes that fit measures its volume: the space inside the solid. A container with a wider base or greater height usually holds more. But the shape of the solid matters too. In this lesson, you will use the relationships between prisms and pyramids, and between cylinders and cones, to find and compare volumes. A base is the face or region at the bottom of a solid. The height is the straight, perpendicular distance from the base to the top. For a shape with a pointed top, the height is measured from the base directly to the point, not along a sloping side.

What you will learn

Describe how the volume of a prism or cylinder relates to its base area and height.
Compare a pyramid with a prism, or a cone with a cylinder, when each pair has the same base and height.
Use these relationships to calculate and compare volumes.
Use cubic units and explain what a calculated volume means.

1. Build the idea from a straight-sided container

A rectangular prism is a box-shaped solid. If its base is

4

centimetres long and

3

centimetres wide, the base covers

12

square centimetres. Each layer of cubes that is

1

centimetre high contains

12

cubes. If the prism is

5

centimetres high, it has five such layers, so it holds

60

cubic centimetres.

This gives a useful way to think about the volume of any prism: find the area of its base, then multiply by its perpendicular height. The base does not have to be a rectangle. It could be another polygon, such as a triangle. The method is the same: use the area of that base and multiply by the height of the prism.

A cylinder has circular bases. It also has the same cross-section all the way from one base to the other, so its volume follows the same relationship. Find the area of its circular base and multiply by its height. The base area is measured in square units; multiplying by a height in units gives cubic units.

A cubic centimetre, written

ext{cm}^3

, is the volume of a cube with edges

1

centimetre long. Keep units consistent: if the base area is in square centimetres and the height is in centimetres, the volume is in cubic centimetres.

V=Bh

For a prism or cylinder, volume is base area multiplied by perpendicular height.
Use the area of the actual base, not the area of a side face.
Volume is written in cubic units, such as $ext{cm}^3$ .

2. Compare a pyramid with a prism

Now picture a prism and a pyramid with matching bases and the same height. Matching bases have the same shape and area. The prism has straight sides from its base to a parallel top. The pyramid narrows from its base to one point.

For a matching pair, the pyramid holds one-third as much as the prism. This relationship works for any pyramid and prism with the same base area and height. It is the relationship you need; you do not need to build or fill the solids to use it.

For example, if a prism has a base area of

30

square centimetres and a height of

9

centimetres, its volume is

270

cubic centimetres. A pyramid with the same base area and height has one-third of that volume, or

90

cubic centimetres. If you know the pyramid’s volume instead, you can multiply by

3

to find the matching prism’s volume.

The letter

B

stands for the area of the base, and

h

stands for the perpendicular height. The pyramid relationship can be written using these quantities. Use the height from the base to the point, not the length of a sloping triangular face.

V_{

\text{pyramid}

}=

\frac{1}{3}

Bh

A pyramid with the same base area and height as a prism has one-third its volume.
To compare in the other direction, a matching prism has three times the pyramid’s volume.
The pyramid’s height is perpendicular to its base.

3. Compare a cone with a cylinder

A cone has a circular base and narrows to a point. A cylinder has a circular base and continues straight to a matching circular top. If a cone and cylinder have the same base area and the same perpendicular height, the cone holds one-third as much as the cylinder.

This is the cone-and-cylinder version of the pyramid-and-prism relationship. First find the area of the circular base. Then multiply by the height for the cylinder. For a cone with that same base area and height, take one-third of the cylinder’s volume.

The radius is the distance from the centre of a circle to its edge. The area of a circular base is found using the circle-area rule. In that rule,

r

means radius and

ext{pi}

is a number used for circles, approximately

3.14

. Use the same base area and height when comparing a cone to a cylinder. Do not compare solids that have different base areas or heights by using the one-third relationship alone.

The table summarizes the two pairs. “Same base and height” means the base areas match and the perpendicular heights match. If either measurement differs, calculate each volume using its own measurements before comparing.

V_{

\text{cone}

}=

\frac{1}{3}

Bh

A cone with the same base area and height as a cylinder has one-third its volume.
For both pairs, the pointed solid has one-third the volume of the matching straight-sided solid.
The circular base area depends on the radius.

4. Choose a relationship and check the result

When solving a volume problem, identify the solid first. Then decide whether the question asks for a prism or cylinder volume, or for a pyramid or cone volume. For a prism or cylinder, multiply base area by height. For a pyramid or cone, use one-third of the product of base area and height.

If the problem gives the base dimensions rather than base area, find the base area first. For a rectangular base, multiply its length by its width. For a circular base, use the circle-area rule. Then use the correct volume relationship. Keep the base-area calculation separate so it is easier to see where each number comes from.

You can check whether an answer is reasonable by comparing it with a matching solid. A pyramid or cone must have less volume than a prism or cylinder with the same base and height. Its volume must be one-third as large. Also check the units: area uses square units, while volume uses cubic units.

A common question asks how many times as large one volume is as another. Divide the larger volume by the smaller volume, or use the one-third relationship when the bases and heights match. State what the comparison means in words, and include units for volume values.

V_{prism or cylinder}=Bh

Identify the solid and the given measurements before choosing a relationship.
Find the base area before using the solid’s volume relationship.
Check that units are cubic and that a matching pointed solid has one-third the volume.

Volume relationships for matching solids

Solid	Volume relationship	Meaning when base area and height match
Prism	$V=Bh$	Reference volume
Pyramid	$V=\frac{1}{3}Bh$	One-third of matching prism
Cylinder	$V=Bh$	Reference volume
Cone	$V=\frac{1}{3}Bh$	One-third of matching cylinder

Worked example

Compare a cone and its matching cylinder

A cylinder and a cone have the same circular base, with radius

3

centimetres, and the same height of

8

centimetres. Find the volume of each solid and compare them. Use

\text{pi}\approx3.14

.

Find the base area
Both solids have the same circular base. Find its area using the radius. This area will be used for both volumes.
$B\approx3.14(3^2)=28.26\text{ cm}^2$
Find the cylinder volume
A cylinder’s volume is its base area multiplied by its height. The result uses cubic centimetres because square centimetres are multiplied by centimetres.
$V_{\text{cylinder}}\approx28.26(8)=226.08\text{ cm}^3$
Find the cone volume
The cone has the same base area and height, so its volume is one-third the cylinder’s volume. Divide the cylinder volume by $3$ .
$V_{\text{cone}}\approx\frac{226.08}{3}=75.36\text{ cm}^3$

Answer: The cylinder’s volume is approximately

226.08\text{ cm}^3

. The cone’s volume is approximately

75.36\text{ cm}^3

. The cylinder holds three times as much as the cone.

Check: Three times

75.36

is

226.08

, so the volumes match the one-third relationship. Both answers are in cubic centimetres.

Common mistakes and how to avoid them

Using the sloping side of a pyramid or cone as its height.

Correction: Use the perpendicular distance from the base to the point. A sloping side is not the height used in the volume relationship.

Finding a pyramid’s or cone’s volume by multiplying base area by height without taking one-third.

Correction: For a pyramid or cone, take one-third of the base area multiplied by the perpendicular height.

Using a base length instead of the whole base area.

Correction: First calculate the area of the base. Then multiply that area by the height, using the one-third relationship when needed.

Writing a volume in square units.

Correction: Volume uses cubic units because it measures three dimensions. Write units such as

\text{cm}^3

.

Lesson summary

A prism or cylinder has volume equal to its base area multiplied by its perpendicular height.
A pyramid or cone with the same base area and height as a matching prism or cylinder has one-third its volume.
Find the base area first, use the correct relationship, and express volume in cubic units.

Check your understanding

Question 1

A prism and pyramid have the same base area and height. The prism’s volume is

150\text{ cm}^3

. What is the pyramid’s volume?

$50\text{ cm}^3$
$75\text{ cm}^3$
$450\text{ cm}^3$
$150\text{ cm}^3$

Show answer and explanation

50\text{ cm}^3

The pyramid has one-third the volume of a matching prism. One-third of

150

is

50

.

Question 2

A cylinder and cone have the same base area and height. Which statement is correct?

The cone has one-third the cylinder’s volume.
The cone has three times the cylinder’s volume.
The two volumes are always equal.
The cylinder has one-third the cone’s volume.

Show answer and explanation

The cone has one-third the cylinder’s volume.

For matching base area and height, the cone’s volume is one-third of the cylinder’s volume.

Question 3

A prism has base area

18\text{ m}^2

and height

4\text{ m}

. What is its volume?

$72\text{ m}^3$
$22\text{ m}^3$
$36\text{ m}^3$
$72\text{ m}^2$

Show answer and explanation

72\text{ m}^3

Multiply base area by height:

18(4)=72

. The volume is in cubic metres.

Key terms

Base: The face or region used as the bottom of a solid when finding its volume.
Base area: The area of the base, measured in square units.
Height: The perpendicular distance from the base to the opposite face or, for a pointed solid, to its point.
Volume: The amount of space inside a solid, measured in cubic units.
Prism: A solid with matching parallel ends and a constant cross-section between them.
Pyramid: A solid with a base and triangular side faces that meet at one point.
Cylinder: A solid with matching circular bases and a constant circular cross-section.
Cone: A solid with a circular base that narrows to one point.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation E1.6. 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.

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E1.6 · Apply volume relationships for prisms, pyramids, cylinders, and cones

Ontario Grade 9 Mathematics

How shape and matching dimensions affect the space a solid holds

What you will learn

1. Build the idea from a straight-sided container

2. Compare a pyramid with a prism

3. Compare a cone with a cylinder

4. Choose a relationship and check the result

Volume relationships for matching solids

Compare a cone and its matching cylinder

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MTH1W

About this lesson and its review