# 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
row 1   $27\,\text{in}^3$
row 2
$\sqrt{5}$

row 3

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

## 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: $$4^2=16$$ $$\sqrt{16}=4$$ 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:

$$4^3=64$$

$$\sqrt{64}=4$$ $\sqrt{64}$ is pronounced “The cube root of 64.” Here are some other values of cube roots:

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

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

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

## Glossary

cube root

#### cube root

The cube root of a number $x$, written $\sqrt{x}$, is the number whose cube is $x$. For example, $\sqrt{8} = 2$ because $2^3 = 8$. The cube root of 0 is 0.