3.3 Surface Area

Leanne Thompson

3.3 Surface Area

Surface area is the sum of the faces of a three-dimensional figure. Similar to perimeter and area, there are many applications of surface area in everyday life. For example, if a company is producing tents, they can figure out how much material is needed for the tent by finding the surface area. Similarly, if you are wrapping a gift, you can figure out how much wrapping paper you will need by finding the surface area of the gift. When calculating surface area, the answer will always be a two-dimensional measurement such as feet squared (or square feet, ft²), inches squared (or square inches, in²), metres squared (or square metres, m²), or centimetres squared (or square centimetres, cm²).

In this lesson, we will cover how to find the surface area of a cube, rectangular prism, and cylinder. The table below outlines what these shapes are.

Table 3.3.1
Shape Name and Definition	Diagram
Cube: A three-dimensional figure made up of six equal squares.
Rectangular prism: A three-dimensional figure made up of six rectangles.
Cylinder: A three-dimensional figure made up of two parallel circular bases joined by a curved surface.

The table below outlines formulas that can be used to help you find the surface area (SA) of these shapes.

Table 3.3.2
Shape	Formula
Cube	$SA=6{s}^{2}$
Rectangular prism	$SA=2lh+2lw+2wh$
Cylinder	$SA=2 \pi r^2+2 \pi rh$

Surface Area of a Cube

Example 1

Find the surface area of a cube with side lengths of 2.5 inches.

Table 3.3.3
Steps	Solution
Draw the figure and label it with the given information.
Choose the appropriate formula.	$SA=6{s}^{2}$
Substitute the values into the formula.	$SA=6 \bullet {\left(2.5\right)}^{2}$
Calculate the surface area. Make sure you include the units with your answer.	$SA=37.5$ $in^2$ The surface area is 37.5 square inches.

Practice 1

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

a) Find the surface area of this cube. The side measurements are in centimetres.

Cube with side lengths of 2.

b) A packing box is a cube measuring 4.5 feet on each side. Find its surface area.

Surface Area of a Rectangular Prism

Example 2

Find the surface area of a rectangular prism with a length of 14 cm, height of 17 cm, and width of 9 cm.

Table 3.3.4
Steps	Solution
Draw the figure and label it with the given information.
Choose the appropriate formula.	$SA=2lh+2lw+2wh$
Substitute the values into the formula.	$SA=2\left(14 \bullet 17\right)+2\left(14 \bullet 9\right)+2\left(9 \bullet 17\right)$
Calculate the surface area. Make sure you include the units with your answer.	$SA=1 034$ $cm^2$ The surface area is 1 034 square centimetres.

Practice 2

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

a) A rectangular crate has a length of 18 mm, width of 12 mm, and height of 5 mm. Find its surface area.

b) Find the surface area of this rectangular prism. The units of the side measurements are in feet.

Rectangular prism with length of 30 width of 25 and height of 20.

Surface Area of a Cylinder

To find the surface area of a cylinder, we can use the formula $SA=2 \pi r^2+2 \pi rh$ . The diagram and work below show how this formula can be derived.

**Table 3.3.5**
A cylinder is made up of two circles and a rectangle:
SA = A_{top circle}+ A_{bottom circle}+ A_rectangle $SA= \pi r^2+\pi r^2+2 \pi rh$ $SA=2 \pi r^2+2 \pi rh$

Example 3

Find the surface area of a cylinder with a height of $6$ cm and a radius of $3$ cm.

Table 3.3.6
Steps	Solution
Draw the figure and label it with the given information.
Choose the appropriate formula.	$SA=2 \pi r^2+2 \pi rh$
Substitute the values into the formula.	$SA\approx 2\left(3.14\right){3}^{2}+2\left(3.14\right)\left(3\right)5$
Calculate the surface area. Make sure you include the units with your answer.	$SA\approx 150.72$ $cm^2$ The surface area is 150.72 square inches.

Practice 3

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

a) Find the surface area of a cylindrical drum with a radius of 2.7 feet and a height of 4 feet. Assume that the drum is shaped exactly like a cylinder.

b) Find the surface area of this can of soda. The measurements on the diagram are in centimetres. Assume the can is shaped exactly like a cylinder.

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

Surface Area Word Problems

Practice 4

A rectangular shipping container has a length of 22.8 feet, width of 8.5 feet, and height of 8.2 feet. If the bottom of the shipping container has the dimensions of 22.8 feet and 8.2 feet, how much paint would you need to paint all of the shipping container excluding the bottom?

Practice 5

A museum is the shape of a cube. Given that the surface area of the museum is 12 150 m², what is the side length of the museum?

Practice 6

A snack pack of cookies is shaped like a cylinder with a radius of 4 cm and a height of 3 cm. The side and bottom are made of cardboard and the lid is made of plastic. How much cardboard is needed to make the snack pack?

Homework

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

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

b) Find the surface area.

c) Find the surface area.

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

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

f) Find the surface area of the cube.
A rectangular moving van has a length of 16 feet, a width of 8 feet, and a height of 8 feet. Find its surface area.
Each side of the cube at the Discovery Science Center in Santa Ana, California, is 64 feet long. Find its surface area.
A can of coffee has a radius of 5 cm and a height of 13 cm. Find its surface area.
A rectangular carton has a length of 21.3 cm, a width of 24.2 cm, and a height of 6.5 cm. Find its surface area.
The base of a statue is a cube with sides that measure 2.8 metres long. Find its surface area.
A cylindrical barbershop pole has a diameter of 6 inches and a height of 24 inches. Find its surface area.
The area of the rectangular portion of a cylinder is 245.04 cm². The height is 10 cm. What is the radius? Round to the nearest tenth.
A classroom is 41 ft long, 25 ft wide, and 16 ft high. The school is painting the four walls and the ceiling of the classroom. Find the total surface area that they are planning to paint.
Vita makes bars of soap with a cylindrical mould. She wraps the bars in paper before she sells them. How much paper would be needed to wrap each bar of soap if the diameter is 1.75 in and the height is 2.5 in? Round to the nearest hundredth.
A die is the shape of a cube. Given that the total surface area of the die is 0.0054 m², what is the side length of the die? (Note: “Die” is the singular of “dice.”)
A soup company is making a label for their cans. If a can has a height of 5.5 in and a diameter of 4 in, how much material does the company need for each soup label? Round to the nearest tenth.

Answers

1.

a) 27 m²

b) 584.8 cm²

c) 406 mm²

d) 63.8 mm²

e) 23.3 m²

f) 486 cm²

2. 640 ft²

3. 24 576 ft²

4. 565.5 cm²

5. 1 622.42 cm²

6. 47.04 m²

7. 508.68 in²

8. 3.9 cm

9. 3 137 ft²

10. 18.56 in²

11. 0.03 m

12. 69.1 in²

Attribution

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

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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