# 1.6.2: Surface Area of a Cube

- Page ID
- 39649

## Lesson

Let's write a formula to find the surface area of a cube.

Exercise \(\PageIndex{1}\): Exponent Review

Select the greater expression of each pair without calculating the value of each expression. Be prepared to explain your choices.

- \(10\cdot 3\) or \(10^{3}\)
- \(13^{2}\) or \(12\cdot 12\)
- \(97+97+97+97+97+97\) or \(5\cdot 97\)

Exercise \(\PageIndex{2}\): The Net of a Cube

- A cube has edge length 5 inches.
- 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?

- A second cube has edge length 17 units.
- 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.

Exercise \(\PageIndex{3}\): 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 \(s^{3}\).

A cube has 6 faces that are all identical squares. The surface area of a cube with edge length \(s\) is \(6\cdot s^{2}\).

### Glossary Entries

Definition: 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 \(s\cdot s\cdot s\), or \(s^{3}\).

Definition: 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, \(5^{3} = 5\cdot 5\cdot 5\), and \(8^{2}=8\cdot 8\).

Definition: 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 \(s\cdot s\), or \(s^{2}\).

## Practice

Exercise \(\PageIndex{4}\)

- What is the volume of a cube with edge length 8 in?
- What is the volume of a cube with edge length \(\frac{1}{3}\) cm?
- A cube has a volume of 8 ft
^{3}. What is its edge length?

Exercise \(\PageIndex{5}\)

- What three-dimensional figure can be assembled from this net?

- If each square has a side length of 61 cm, write an expression for the surface area and another for the volume of the figure.

Exercise \(\PageIndex{6}\)

- Draw a net for a cube with edge length \(x\) cm.
- What is the surface area of this cube?
- What is the volume of this cube?

Exercise \(\PageIndex{7}\)

Here is a net for a rectangular prism that was not drawn accurately.

- Explain what is wrong with the net.
- Draw a net that can be assembled into a rectangular prism.
- Create another net for the same prism.

(From Unit 1.5.3)

Exercise \(\PageIndex{8}\)

State whether each figure is a polyhedron. Explain how you know.

(From Unit 1.5.2)

Exercise \(\PageIndex{9}\)

Here is Elena’s work for finding the surface area of a rectangular prism that is 1 foot by 1 foot by 2 feet.

She concluded that the surface area of the prism is 296 square feet. Do you agree with her? Explain your reasoning.

(From Unit 1.5.1)