3.4 Volume

Leanne Thompson

3.4 Volume

Volume is the amount of space that a three-dimensional figure takes up. There are many applications of volume in everyday life. For example, if you are filling a swimming pool, you can figure out how much water is needed by finding the volume of the swimming pool. Similarly, if a company is producing boxes of cereal, they can figure out how much cereal can fit in a box by finding the volume. When calculating volume, the answer will always be a three-dimensional measurement such as feet cubed (or cubic feet, ft³), inches cubed (or cubic inches, in³), metres cubed (or cubic metres, m³), or centimetres cubed (or cubic centimetres, cm³).

In this lesson, we will cover how to find the volume (V) of a variety of right prisms and a cylinder. The table below outlines formulas that can be used to help you find the volume of these shapes.

Table 3.4.1
Shape	Formula
All right prisms	$V=A_{base}\times h$
Cylinder	$V= \pi r^2 h$

Volume of a Right Prism

A right prism is a three-dimensional figure made up of two identical faces connected by rectangular faces. Here are some examples of right prisms:

Table 3.4.2
Rectangular Prism	Triangular Prism	Cube

To find volume of any right prism, find the area of the base, and then multiply it by the height of the prism.

Example 1

Find the volume of the given triangular prism.

Triangular prism base of 3 cm, triangle height of 4 cm, and prism height of 12 cm.

Table 3.4.3
Steps	Solution
Choose the appropriate formula.	$V=A_{base}\times h$
Change the formula to reflect the base of the figure you are working with.	$V=\frac{bh}{2}\times H$
Substitute the values into the formula.	$V=\frac{3 \times 4}{2}\times 12$
Calculate the volume. Make sure you include the units with your answer.	$V=72$ $cm^3$ The volume is 72 cubic centimetres.

Practice 1

Answer the following questions. Round to the nearest tenth where necessary.

a) Find the volume of this figure. The units of the side measurements are inches.

b) Find the volume of this figure. The units of the side measurements are metres.

c) A triangular prism has a height of 3.23 yd. The triangular bases have a height of 2.11 yd and base of 3.98 yd. Find the volume of this figure.

d) A rectangular gift box has a length of 26 inches, width of 16 inches, and height of 4 inches. Find its volume.

Volume of a Cylinder

Example 2

A cylinder has a height of 5 centimetres and a radius of 3 centimetres. Find the volume.

Table 3.4.4
Steps	Solution
Draw the figure and label it with the given information.
Choose the appropriate formula.	$V=\pi {r}^{2}h$
Substitute the values into the formula.	$V= \left(\pi \right){3}^{2} \bullet 5$
Calculate the volume. Make sure you include the units with your answer.	$V= 141.3$ $cm^3$ The volume is 141.3 cubic centimetres.

Practice 2

Answer the following questions. Round to the nearest hundredth where necessary.

a) Find the volume of a cylinder with a radius of 2 ft and a height of 8 ft. Draw a diagram.

b) Find how much soda can fit in this can of soda. Assume that the can is shaped exactly like a cylinder.

Cylindrical can of soda with a height of 13 and a radius of 4.

Volume Word Problems

Practice 3

A box of cat food holds 601.152 in³ of food. If the width of the box is 3.1 in and the height is 20.2 in, what is the third dimension of the box ?

Practice 4

Does a cube with sides measuring 8 ft or a cylinder with a height of 9 ft and a diameter of 8 ft have a larger volume? Explain your reasoning.

Practice 5

A box of tissues is a cube with a volume of 9.125 in³. What is the length of each side of the box?

Homework

Answer the following questions. Round to the nearest hundredth where necessary.

a) A rectangular prism has a length of 2 metres, width of 1.5 metres, and height of 3 metres. Find its volume.

b) Find the volume.

cylinder with radius of 5.2 cm and height of 12.7 cm

c) Find the volume.

rectangular prism with side lengths 14 mm 5 mm and 7 mm

d) A cube measures 3.26 mm on each side. Find its volume.

e) Find the volume of a cylindrical prism with a diameter of 1.5 metres and a height of 4.2 metres.

f) Find the volume.

cube with side length 9.0 cm

g) A triangular prism has a height of 18 cm. The triangular bases have a height of 5 cm and a base of 6 cm. Find the volume of this figure.

h) Find the volume of the given triangular prism.

Triangular prism base of 0.21 m, triangle height of 65 cm, and prism height of 23 cm.

A rectangular moving van has a length 16 feet, width of 8 feet, and height of 8 feet. Find its volume. Round to the nearest hundredth, if necessary.
Each side of the cube at the Discovery Science Center in Santa Ana is 64 feet long. Find its volume. Round to the nearest hundredth, if necessary.
A can of coffee has a radius of 5 cm and a height of 13 cm. Find its volume. Round to the nearest hundredth, if necessary.
A rectangular carton has length 21.3 cm, width 24.2 cm, and height 6.5 cm. Find its volume. Round to the nearest hundredth, if necessary.
The base of a statue is a cube with sides 2.8 metres long. Find its volume. Round to the nearest hundredth, if necessary.
A cylindrical barbershop pole has a diameter of 6 inches and a height of 24 inches. Find its volume. Round to the nearest hundredth, if necessary.
A triangular prism has a base that measures 6 cm. The height of the triangle measures 4 cm, and the height of the prism measures 10 cm. Find the volume. Round to the nearest hundredth, if necessary.

<
div class=”question-block”>A ramp for a skate park is in the shape of a triangular prism. The base of the triangle is 8 metres, and the height of the triangle is 4 metres. The length of the ramp (the height of the prism) is 15 metres. What is the volume of the ramp? Round to the nearest hundredth, if necessary.
A storage tank shaped like a triangular prism has a volume of 1 200 cubic metres. The base of the triangular cross-section is 10 metres, and the height of the triangle is 8 metres. Find the length of the tank. Round to the nearest hundredth, if necessary.
The volume of a cube-shaped storage box is 512 cubic centimetres. Find the length of one side of the cube. Round to the nearest hundredth, if necessary.
A gift box in the shape of a rectangular prism has a volume of 720 cubic centimetres. The length of the box is 18 cm, and the width is 12 cm. Find the height of the box. Round to the nearest hundredth, if necessary.

Answers

1.

a) 9 m³	b) 1 078.85 cm³	c) 490 mm³	d) 34.65 mm³
e) 7.42 m³	f) 729 cm³	g) 270 cm³	h) 15 697.5 cm³

2. 1 024 ft³

3. 262 144 ft³

4. 1 021.02 cm³

5. 3 350.49 cm³

6. 21.952 m³

7. 678.58 in³

8. 120 cm³

9. 240 m³

10. 30 m

11. 8 cm

12. 3.33 cm

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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