# Lesson 12

Edge Lengths and Volumes

Let’s explore the relationship between volume and edge lengths of cubes.

### 12.1: Ordering Squares and Cubes

Let $$a$$, $$b$$, $$c$$, $$d$$, $$e$$, and $$f$$ be positive numbers.

Given these equations, arrange $$a$$, $$b$$, $$c$$, $$d$$, $$e$$, and $$f$$ from least to greatest. Explain your reasoning.

• $$a^2 = 9$$

• $$b^3 = 8$$

• $$c^2 = 10$$

• $$d^3 = 9$$

• $$e^2 = 8$$

• $$f^3 = 7$$

### 12.2: Name That Edge Length!

Fill in the missing values using the information provided:

sides volume volume equation
$$27\,\text{in}^3$$

$$\sqrt[3]{5}$$

$$(\sqrt[3]{16})^3=16$$

A cube has a volume of 8 cubic centimeters. A square has the same value for its area as the value for the surface area of the cube. How long is each side of the square?

### 12.3: Card Sort: Rooted in the Number Line

Your teacher will give your group a set of cards. For each card with a letter and value, find the two other cards that match. One shows the location on a number line where the value exists, and the other shows an equation that the value satisfies. Be prepared to explain your reasoning.

### Summary

To review, the side length of the square is the square root of its area. In this diagram, the square has an area of 16 units and a side length of 4 units.

These equations are both true: $$\displaystyle 4^2=16$$ $$\displaystyle \sqrt{16}=4$$

Now think about a solid cube. The cube has a volume, and the edge length of the cube is called the cube root of its volume. In this diagram, the cube has a volume of 64 units and an edge length of 4 units:

These equations are both true:

$$\displaystyle 4^3=64$$

$$\displaystyle \sqrt[3]{64}=4$$

$$\sqrt[3]{64}$$ is pronounced “The cube root of 64.” Here are some other values of cube roots:

$$\sqrt[3]{8}=2$$, because $$2^3=8$$

$$\sqrt[3]{27}=3$$, because $$3^3=27$$

$$\sqrt[3]{125}=5$$, because $$5^3=125$$

### Glossary Entries

• cube root

The cube root of a number $$n$$ is the number whose cube is $$n$$. It is also the edge length of a cube with a volume of $$n$$. We write the cube root of $$n$$ as $$\sqrt[3]{n}$$.

For example, the cube root of 64, written as $$\sqrt[3]{64}$$, is 4 because $$4^3$$ is 64. $$\sqrt[3]{64}$$ is also the edge length of a cube that has a volume of 64.