Lesson 12
Surface Area of a Cube
Let’s write a formula to find the surface area of a cube.
12.1: Exponent Review
Select the greater expression of each pair without calculating the value of each expression. Be prepared to explain your choices.
- \(10 \boldcdot 3\) or \(10^3\)
- \(13^2\) or \(12 \boldcdot 12\)
- \(97+97+97+97+97+97\) or \(5 \boldcdot 97\)
12.2: The Net of a Cube
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A cube has edge length 5 inches.
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Draw a net for this cube, and label its sides with measurements.
- What is the shape of each face?
- What is the area of each face?
- What is the surface area of this cube?
- What is the volume of this cube?
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A second cube has edge length 17 units.
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Draw a net for this cube, and label its sides with measurements.
- Explain why the area of each face of this cube is \(17^2\) square units.
- Write an expression for the surface area, in square units.
- Write an expression for the volume, in cubic units.
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12.3: Every Cube in the Whole World
A cube has edge length \(s\).
- Draw a net for the cube.
- Write an expression for the area of each face. Label each face with its area.
- Write an expression for the surface area.
- Write an expression for the volume.
Summary
The volume of a cube with edge length \(s\) is \(s^3\).
![cube with transparant faces, side length s](https://staging-cms-im.s3.amazonaws.com/YaS3vaDtMo7v1AzH47YAf6ZD?response-content-disposition=inline%3B%20filename%3D%226-6.1.F2_Image_3.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.F2_Image_3.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T115639Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a6503c820bbf5f8a0a6ad85b0b28f3389a14c446a75c3e6aea8bce0f35ff5b78)
A cube has 6 faces that are all identical squares. The surface area of a cube with edge length \(s\) is \(6 \boldcdot s^2\).
![A net for a cube with each square labeled \(s^2\).](https://staging-cms-im.s3.amazonaws.com/MaqnahV1gGHsjum9wXyPaAxa?response-content-disposition=inline%3B%20filename%3D%226-6.1.F2_Image_2.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.F2_Image_2.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T115639Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9905e1762aeaeff427acf3db982c99ad5cafff44a0528aef48757ccd4855694a)
Glossary Entries
- cubed
We use the word cubed to mean “to the third power.” This is because a cube with side length \(s\) has a volume of \(s \boldcdot s \boldcdot s\), or \(s^3\).
- exponent
In expressions like \(5^3\) and \(8^2\), the 3 and the 2 are called exponents. They tell you how many factors to multiply. For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).
- squared
We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).