Lesson 12Edge Lengths and Volumes

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

Learning Targets:

  • I can approximate cube roots.
  • I know what a cube root is.
  • I understand the meaning of expressions like \sqrt[3]{5} .

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:

A cube.
sides volume volume equation
27\,\text{in}^3
\sqrt[3]{5}
(\sqrt[3]{16})^3=16

Are you ready for more?

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.

Lesson 12 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

A square with a side length of 4 units on a square grid.

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[3]{64}=4

A solid cube composed of 64 unit cubes. Each edge length is 4 unit cubes.

\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 Terms

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. 

Lesson 12 Practice Problems

    1. What is the volume of a cube with a side length of
      1. 4 centimeters?
      2. \sqrt[3]{11} feet?
      3. s units?
    2. What is the side length of a cube with a volume of
      1. 1,000 cubic centimeters?
      2. 23 cubic inches?
      3. v cubic units?
  1. Write an equivalent expression that doesn’t use a cube root symbol.

    1. \sqrt[3]{1}
    2. \sqrt[3]{216}
    3. \sqrt[3]{8000}
    4. \sqrt[3]{\frac{1}{64}}
    5. \sqrt[3]{\frac{27}{125}}
    6. \sqrt[3]{0.027}
    7. \sqrt[3]{0.000125}
  2. Find the distance between each pair of points. If you get stuck, try plotting the points on graph paper.

    1. X=(5,0) and Y=(\text-4,0)
    2. K=(\text-21,\text-29) and L=(0,0)
  3. Here is a 15-by-8 rectangle divided into triangles. Is the shaded triangle a right triangle? Explain or show your reasoning.

    A rectangle with a point on the bottom side. Two line segments are drawn from the point to the top left vertex and from the point to the top right vertex of the rectangle creating 3 triangles. The left side of the rectangle is labeled 8. The segment from the bottom left corner of the rectangle to the point on the bottom side is labeled 9. The segment from the point on the bottom side to the bottom right corner is labeled 6. The middle triangle is shaded.
  4. Here is an equilateral triangle. The length of each side is 2 units. A height is drawn. In an equilateral triangle, the height divides the opposite side into two pieces of equal length.

    1. Find the exact height.
    2. Find the area of the equilateral triangle.
    3. (Challenge) Using x for the length of each side in an equilateral triangle, express its area in terms of x .