# 1.6.1: Squares and Cubes

- Page ID
- 39648

## Lesson

Let's investigate perfect squares and perfect cubes.

Exercise \(\PageIndex{1}\): Perfect Squares

- The number 9 is a perfect
**square**. Find four numbers that are perfect squares and two numbers that are not perfect squares. - A square has side length 7 in. What is its area?
- The area of a square is 64 sq cm. What is its side length?

Exercise \(\PageIndex{2}\): Building with 32 Cubes

Use the 32 snap cubes in the applet’s hidden stack to build the largest single cube you can. Each small cube has side length of 1 unit.

- How many snap cubes did you use?
- What is the side length of the cube you built?
- What is the area of each face of the built cube? Show your reasoning.
- What is the volume of the built cube? Show your reasoning.

Are you ready for more?

This applet has a total of 64 snap cubes. Build the largest single cube you can.

GeoGebra Applet XFx3bD7h

- How many snap cubes did you use?
- What is the edge length of the new cube you built?
- What is the area of each face of this built cube? Show your reasoning.
- What is the volume of this built cube? Show your reasoning.

Exercise \(\PageIndex{3}\): Perfect Cubes

- The number 27 is a perfect
**cube**. Find four other numbers that are perfect cubes and two numbers that are*not*perfect cubes. - A cube has side length 4 cm. What is its volume?
- A cube has side length 10 inches. What is its volume?
- A cube has side length units. What is its volume?

Exercise \(\PageIndex{4}\): Introducing Exponents

Make sure to include correct units of measure as part of each answer.

- A square has side length 10 cm. Use an
**exponent**to express its area. - The area of a square is \(7^{2}\) sq in. What is its side length?
- The area of a square is 81 m
^{2}. Use an exponent to express this area. - A cube has edge length 5 in. Use an exponent to express its volume.
- The volume of a cube is \(6^{3}\) cm
^{3}. What is its edge length? - A cube has edge length \(s\) units. Use an exponent to write an expression for its volume.

Are you ready for more?

The number 15,625 is both a perfect square and a perfect cube. It is a perfect square because it equals \(125^{2}\). It is also a perfect cube because it equals \(25^{3}\). Find another number that is both a perfect square and a perfect cube. How many of these can you find?

### Summary

When we multiply two of the same numbers together, such as \(5\cdot 5\), we say we are **squaring** the number. We can write it like this: \(5^{2}\)

Because \(5\cdot 5=25\), we write \(5^{2}=25\) and we say, “5 squared is 25.”

When we multiply three of the same numbers together, such as \(4\cdot 4\cdot 4\), we say we are **cubing** the number. We can write it like this: \(4^{3}\)

Because \(4\cdot 4\cdot 4\cdot =64\), we write \(4^{3}=64\) and we say, “4 cubed is 64.”

We also use this notation for square and cubic units.

- A square with side length 5 inches has area 25 in
^{2}. - A cube with edge length 4 cm has volume 64 cm
^{3}.

To read 25 in^{2}, we say “25 square inches,” just like before.

The area of a square with side length 7 kilometers is \(7\) km^{2}. The volume of a cube with edge length 2 millimeters is \(2^{3}\) mm^{3}.

In general, the area of a square with side length \(s\) is \(s^{2}\), and the volume of a cube with edge length \(s\) is \(s^{3}\).

### 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 \(\PageIndex{5}\)

What is the volume of this cube?

Exercise \(\PageIndex{6}\)

- Decide if each number on the list is a perfect square.
- 16
- 20
- 25
- 100
- 125
- 144
- 225
- 10,000

- Write a sentence that explains your reasoning.

Exercise \(\PageIndex{7}\)

- Decide if each number on the list is a perfect cube.
- 1
- 3
- 8
- 9
- 27
- 64
- 100
- 125

- Explain what a perfect cube is.

Exercise \(\PageIndex{8}\)

- A square has side length 4 cm. What is its area?
- The area of a square is 49 m
^{2}. What is its side length? - A cube has edge length 3 in. What is its volume?

Exercise \(\PageIndex{9}\)

Prism A and Prism B are rectangular prisms.

- Prism A is 3 inches by 2 inches by 1 inch.
- Prism B is 1 inch by 1 inch by 6 inches.

Select **all** statements that are true about the two prisms.

- They have the same volume.
- They have the same number of faces.
- More inch cubes can be packed into Prism A than into Prism B.
- The two prisms have the same surface area.
- The surface area of Prism B is greater than that of Prism A.

(From Unit 1.5.5)

Exercise \(\PageIndex{10}\)

- What polyhedron can be assembled from this net?

- What information would you need to find its surface area? Be specific, and label the diagram as needed.

(From Unit 1.5.3)

Exercise \(\PageIndex{11}\)

Find the surface area of this triangular prism. All measurements are in meters.

(From Unit 1.5.4)