Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1.6.2: Surface Area of a Cube

  • Page ID
    39649
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Lesson

    Let's write a formula to find the surface area of a cube.

    Exercise \(\PageIndex{1}\): Exponent Review

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

    • \(10\cdot 3\) or \(10^{3}\)
    • \(13^{2}\) or \(12\cdot 12\)
    • \(97+97+97+97+97+97\) or \(5\cdot 97\)

    Exercise \(\PageIndex{2}\): 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 second 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 \(17^{2}\) square units.
      3. Write an expression for the surface area, in square units.
      4. Write an expression for the volume, in cubic units.

    Exercise \(\PageIndex{3}\): 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.

    Summary

    The volume of a cube with edge length \(s\) is \(s^{3}\).

    clipboard_e642682b119c3cc6a6463ea8423051570.png
    Figure \(\PageIndex{1}\)

    A cube has 6 faces that are all identical squares. The surface area of a cube with edge length \(s\) is \(6\cdot s^{2}\).

    clipboard_eece80acac99928c74749a0db498f4836.png
    Figure \(\PageIndex{2}\)

    Glossary Entries

    Definition: Cubed

    We use the word cubed to mean “to the third power.” This is because a cube with side length \(s\) has a volume of \(s\cdot s\cdot s\), or \(s^{3}\).

    Definition: Exponent

    In expressions like \(5^{3}\) and \(8^{2}\), the 3 and the 2 are called exponents. They tell you how many factors to multiply. For example, \(5^{3} = 5\cdot 5\cdot 5\), and \(8^{2}=8\cdot 8\).

    Definition: Squared

    We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s\cdot s\), or \(s^{2}\).

    Practice

    Exercise \(\PageIndex{4}\)

    1. What is the volume of a cube with edge length 8 in?
    2. What is the volume of a cube with edge length \(\frac{1}{3}\) cm?
    3. A cube has a volume of 8 ft3. What is its edge length?

    Exercise \(\PageIndex{5}\)

    1. What three-dimensional figure can be assembled from this net?
    clipboard_eb5f58006c4fe798f05c2036b68a78077.png
    Figure \(\PageIndex{3}\)
    1. 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.

    Exercise \(\PageIndex{6}\)

    1. Draw a net for a cube with edge length \(x\) cm.
    2. What is the surface area of this cube?
    3. What is the volume of this cube?

    Exercise \(\PageIndex{7}\)

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

    clipboard_ebd12e556c6622dcd9f4922d4117a4249.png
    Figure \(\PageIndex{4}\)
    1. Explain what is wrong with the net.
    2. Draw a net that can be assembled into a rectangular prism.
    3. Create another net for the same prism.

    (From Unit 1.5.3)

    Exercise \(\PageIndex{8}\)

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

    clipboard_ecf3dd54757b546d0bda9bb3770ca3600.png
    Figure \(\PageIndex{5}\)

    (From Unit 1.5.2)

    Exercise \(\PageIndex{9}\)

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

    clipboard_ef5c382ac374f5b3f0e7792a78f69bcfc.png
    Figure \(\PageIndex{6}\): Rectangular prism. written top & bottom: 2 times 12 times 12 = 2 times 144 = 288. four side faces: 4 times 2 times 1 = 8. top face 12 inches by 12 inches. Bottom face 1 foot by 1 foot. height 2 feet.

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

    (From Unit 1.5.1)