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Illustrative Mathematics Unit 6.1, Lesson 18: Squares and Cubes

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Learn how to write and explain the formula for the surface area of a cube. After trying the questions, click on the buttons to view answers and explanations in text or video.

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Surface Area of a Cube
Let’s write a formula to find the surface area of a cube.

18.1 - Exponent Review

Select the greater expression of each pair without calculating the value of each expression. Be prepared to explain your choices.

a. 10 · 3 or 103
b. 132 or 12 · 12
c. 97 + 97 + 97 + 97 + 97 + 97 or 5 · 97





18.2 - The Net of a Cube

1. A cube has edge length 5 inches.

  1. Draw a net for this cube, and label its sides with measurements.
  2. What is the shape of each face?
  3. What is the area of each face?
  4. What is the surface area of this cube?
  5. What is the volume of this cube?

2. A cube has edge length 17 units.

  1. Draw a net for this cube, and label its sides with measurements.
  2. Explain why the area of each face of this cube is 172 square units.
  3. Write an expression for the surface area, in square units.
  4. Write an expression for the volume, in cubic units.
  • See Possible Answers

    Two figures: a net of a cube with edge length 5 inches, and a net of a cube with edge length 17 units.

    1. b. Square faces
    c. Area of one face = 5 in × 5 in = 25 in2
    d. Surface area = 6 faces × 25 in2 = 150 in2
    e. Volume = 5 in × 5 in × 5 in = 53 = 125 in3

    2. b. Area of one face (area formula of a square) = 17 in × 17 in = 172 in2
    c. Surface area = 6 faces × 172 in2
    d. Volume = 17 in × 17 in × 17 in = 173




18.3 - Every Cube in the Whole World

A cube has edge length s.

1. Draw a net for the cube.

2. Write an expression for the area of each face. Label each face with its area.

3. Write an expression for the surface area.

4. Write an expression for the volume.

  • See Possible Answers

    A net of a cube with edge length s and face area s squared.

    2. Area of one face (area formula of a square) = s · s = s2
    The area of each face is underlined.

    3. Surface area = 6 faces · s2

    4. Volume = s · s · s = s3



Lesson 18 Summary

A cube with edge length s.

The volume of a cube with edge length s is s3.

A cube has 6 faces that are all identical squares. The surface area of a cube with edge length s is 6 · s2.

A net for a cube with each square labeled $s^2$.



Practice Problems

1. a. What is the volume of a cube with edge length 8 in?
b. What is the volume of a cube with edge length ⅓ cm?
c. A cube has a volume of 8 ft3. What is its edge length?

  • Answers

    a. 83 in3 = 512 in3

    b. (⅓)3 cm3 = 127 cm3

    c. 23 ft3 = 8 ft3
    2 ft is the edge length.


2. a. What three-dimensional figure can be assembled from this net?

An unlabelled net consisting of 4 squares in a row, with 1 square above and 1 square below the 2nd square for a total of 6 squares.

b. If each square has a side length of 61 cm, write an expression for the surface area and another for the volume of the figure.

  • Answers

    a. This net can be assembled into a cube.

    b. Surface area = 6(612) cm2
    Volume = 613 cm3


3. a. Draw a net for a cube with edge length x cm.
b. What is the surface area of this cube?
c. What is the volume of this cube?

  • Answers

    a.
    A net of a cube with edge length x.

    b. Surface area = 6 faces · x2

    c. Volume = x · x · x = x3


4. Here is a net for a rectangular prism that was not drawn accurately.

A net consisting of 4 rectangles in a row with a square above the 2nd rectangle and another square below the 4th rectangle.

a. Explain what is wrong with the net.
b. Draw a net that can be assembled into a rectangular prism.
c. Create another net for the same prism.

  • See Possible Answers

    a. The square faces are too small to match the entire edge of the large rectangles when the net is folded.

    b and c.
    Two nets. Left: 4 rectangles in a row with a rectangle above the 2nd rectangle, and another rectangle below the 4th rectangle, for a total of 6 rectangles. Right: 4 rectangles in a row with a rectangle above and below the 2nd rectangle, for a total of 6 rectangles.


5. State whether each figure is a polyhedron. Explain how you know.

Two figures labeled A and B. A is a cylinder and B is a heptagonal prism.

  • Answers

    A is not a polyhedron, while B is a polyhedron. A is a cylinder, which has non-polygonal faces. B is a heptagonal prism and all its faces are polygons.


6. Here is Elena’s work for finding the surface area of a rectangular prism that is 1 foot by 1 foot by 2 feet.

A hand-drawn sketch of a rectangular prism 1 foot by 1 foot by 2 feet. The top face is labeled 12 in by 12 in, the bottom face is labeled 1 ft by 1 ft, and the visible side faces are labeled 2 by 1 ft. The sketch states that the top and bottom faces have a total area of 2 times 12 times 12 = 288, and that the four side faces have a total area of 4 times 2 times 1 = 8.

She concluded that the surface area of the prism is 296 square feet. Do you agree with her conclusion? Explain your reasoning.

  • Answers

    Elena is incorrect. The dimensions of the top face have been labeled in inches, which is a different unit from the rest of the labels. Values calculated in square inches (the total area of the top and bottom faces in Elena's sketch and calculations = 288 in2) cannot be added to values calculated in square feet (the total area of the 4 side faces in Elena's sketch and calculations = 8 ft2).



The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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