# Lesson 11

Squares and Cubes

Let’s investigate perfect squares and perfect cubes.

### 11.1: 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?

### 11.2: 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.

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

- 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.

### 11.3: 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 \(s\) units. What is its volume?

### 11.4: 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.

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 \(5\boldcdot 5\), we say we are **squaring** the number. We can write it like this: \(\displaystyle 5^2\)

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

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

Because \(4\boldcdot 4\boldcdot 4 = 64\), 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 \(7^2\) 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

**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 \(s^2\).