6.2: Surface Area of a Cube
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
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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?
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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)