Lesson 11Squares and Cubes

Learning Goal

Let’s investigate perfect squares and perfect cubes.

Learning Targets

  • I can write and explain the formula for the volume of a cube, including the meaning of the exponent.

  • When I know the edge length of a cube, I can find the volume and express it using appropriate units.

Lesson Terms

  • cubed
  • exponent
  • squared

Warm Up: Perfect Squares

Problem 1

The number 9 is a perfect square.

A blue square on a blank grid with sides of 3.

  1. Find four numbers that are perfect squares and two numbers that are not perfect squares.

  2. A square has side length 7 km. What is its area?

  3. The area of a square is 64 sq cm. What is its side length?

Activity 1: Building with 32 Cubes

Problem 1

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.

open applet in presentation mode

  1. How many snap cubes did you use?

  2. What is the side length of the cube you built?

  3. What is the area of each face of the built cube? Show your reasoning.

  4. What is the volume of the built cube? Show your reasoning.

Print Version

Your teacher will give you 32 snap cubes. Use them to build the largest single cube you can. Each small cube has an edge length of 1 unit.

  1. How many snap cubes did you use?

  2. What is the edge length of the cube you built?

  3. What is the area of each face of the built cube? Show your reasoning.

  4. What is the volume of the built cube? Show your reasoning.

Are you ready for more?

Problem 1

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

open applet in presentation mode

  1. How many snap cubes did you use?  

  2. What is the edge length of the new cube you built?

  3. What is the area of each face of this built cube? Show your reasoning.  

  4. What is the volume of this built cube? Show your reasoning.

Print Version

Your teacher will give you 64 snap cubes. Use them to build the largest single cube you can. Each small cube has an edge length of 1 unit.

  1. How many snap cubes did you use?

  2. What is the edge length of the new cube you built?

  3. What is the area of each face of this built cube? Show your reasoning.  

  4. What is the volume of this built cube? Show your reasoning.

Activity 2: Perfect Cubes

Problem 1

The number 27 is a perfect cube.

A 3 by 3 by 3 cube.

  1. Find four other numbers that are perfect cubes and two numbers that are not perfect cubes.

  2. A cube has side length 4 cm. What is its volume?

  3. A cube has side length 10 inches. What is its volume?

  4. A cube has side length units. What is its volume?

Activity 3: Introducing Exponents

Problem 1

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

  1. A square has side length 10 cm. Use an exponent to express its area.

  2. The area of a square is sq in. What is its side length?

  3. The area of a square is 81 m². Use an exponent to express this area.

  4. A cube has edge length 5 in. Use an exponent to express its volume.

  5. The volume of a cube is cm³. What is its edge length?

  6. A cube has edge length units. Use an exponent to write an expression for its volume.

Are you ready for more?

Problem 1

The number is both a perfect square and a perfect cube. It is a perfect square because it equals . It is also a perfect cube because it equals . Find another number that is both a perfect square and a perfect cube. How many of these can you find?

Lesson Summary

When we multiply two of the same numbers together, such as , we say we are squaring the number. We can write it like this: Because , we write and we say, “ squared is .”

When we multiply three of the same numbers together, such as , we say we are cubing the number. We can write it like this: Because , we write 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 in².

  • A cube with edge length 4 cm has volume 64 cm³.

To read 25 in², we say “25 square inches,” just like before.

The area of a square with side length 7 kilometers is km². The volume of a cube with edge length 2 millimeters is mm³.

In general, the area of a square with side length is , and the volume of a cube with edge length is .