Finding the Volume and Surface Area of a Cube

Learning Outcomes

  • Find the volume and surface area of a cube

 

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

[latex]\begin{array}{ccccc}V=LWH\hfill & & & & S=2LH+2LW+2WH\hfill \\ V=s\cdot s\cdot s\hfill & & & & S=2s\cdot s+2s\cdot s+2s\cdot s\hfill \\ V={s}^{3}\hfill & & & & S=2{s}^{2}+2{s}^{2}+2{s}^{2}\hfill \\ & & & & S=6{s}^{2}\hfill \end{array}[/latex]
So for a cube, the formulas for volume and surface area are [latex]V={s}^{3}[/latex] and [latex]S=6{s}^{2}[/latex].

Volume and Surface Area of a Cube

For any cube with sides of length [latex]s[/latex],

An image of a cube is shown. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.

 

example

A cube is [latex]2.5[/latex] inches on each side. Find its 1. volume and 2. surface area.

Solution
Step 1 is the same for both 1. and 2., so we will show it just once.

Step 1. Read the problem. Draw the figure and

label it with the given information.

.
1.
Step 2. Identify what you are looking for. the volume of the cube
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.

Write the appropriate formula.

[latex]V={s}^{3}[/latex]
Step 5. Solve. Substitute and solve. [latex]V={\left(2.5\right)}^{3}[/latex]

[latex]V=15.625[/latex]

Step 6. Check: Check your work.
Step 7. Answer the question. The volume is [latex]15.625[/latex] cubic inches.
2.
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.

Write the appropriate formula.

[latex]S=6{s}^{2}[/latex]
Step 5. Solve. Substitute and solve. [latex]S=6\cdot {\left(2.5\right)}^{2}[/latex]

[latex]S=37.5[/latex]

Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is [latex]37.5[/latex] square inches.

 

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example

A notepad cube measures [latex]2[/latex] inches on each side. Find its 1. volume and 2. surface area.

 

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