How to Find Volume and Surface Area of Cubes? (+FREE Worksheet!)

A cube is one of the most perfectly symmetrical three-dimensional shapes in geometry — all six faces are identical squares, all edges are equal in length, and every angle is 90°. On the GED Math test you may need to find the volume (how much space it holds) or the surface area (the total area of all six faces). Both formulas use only one measurement: the side length s.

What Is a Cube?

A cube is a special rectangular prism where \(\color{blue}{\text{ length } = \text{ width } = \text{ height }}\) = s. Every face is a congruent square with area s², and there are six faces. Common real-world cubes include dice, sugar cubes, and Rubik’s cubes.

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Formulas for a Cube

Volume of a Cube

Volume measures the amount of space inside the cube, expressed in cubic units.

\(\color{blue}{V = s^{3}}\)

• Cube with \(\color{blue}{s = 5}\) cm: \(\color{blue}{V = 5^{3} = 5 \times 5 \times 5}\) = 125 cm³
• Cube with \(\color{blue}{s = 3}\) in: \(\color{blue}{V = 3^{3} = 3 \times 3 \times 3}\) = 27 in³

Surface Area of a Cube

Surface area is the total area of all 6 square faces, expressed in square units.

\(\color{blue}{\text{ SA } = 6s^{2}}\)

• Cube with \(\color{blue}{s = 5}\) cm: \(\color{blue}{\text{ SA } = 6 \times 5^{2} = 6 \times 25}\) = 150 cm²
• Cube with \(\color{blue}{s = 7}\) ft: \(\color{blue}{\text{ SA } = 6 \times 7^{2} = 6 \times 49}\) = 294 ft²

Step-by-Step Summary

1. Identify the side length s from the problem or diagram.
2. For volume: cube the side length — \(\color{blue}{V = s^{3}}\).
3. For surface area: square the side length, then multiply by 6 — \(\color{blue}{\text{ SA } = 6s^{2}}\).
4. Label the answer with cubic units (for volume) or square units (for surface area).

Watch: Volume and Surface Area of a Cube (Video Lesson)

This video lesson explains both formulas with clear visual examples:

Worked Examples

Example 1: Find the volume and surface area of a cube with side length 4 m.

\(\color{blue}{V = 4^{3} = 64}\) m³
\(\color{blue}{\text{ SA } = 6 \times 4^{2} = 6 \times 16}\) = 96 m²

Example 2: A cube has a surface area of 54 ft². Find the side length and volume.

\(\color{blue}{\text{ SA } = 6s^{2} = 54}\)  ⇒  \(\color{blue}{s^{2} = 9}\)  ⇒  \(\color{blue}{s = 3}\) ft
\(\color{blue}{V = 3^{3}}\) = 27 ft³

Example 3: How many cubes with \(\color{blue}{s = 2}\) cm can fit inside a cube with \(\color{blue}{s = 6}\) cm?

Volume of large \(\color{blue}{\text{ cube } = 6^{3} = 216}\) cm³
Volume of small \(\color{blue}{\text{ cube } = 2^{3} = 8}\) cm³
\(\color{blue}{\text{ Number } = 216 \div 8}\) = 27 cubes

Example 4: A cubic storage unit has side length 9 ft. How much space does it hold?

\(\color{blue}{V = 9^{3} = 9 \times 9 \times 9}\) = 729 ft³

More Practice: Volume and Surface Area (Video)

Math with Mr. J covers volume of 3-D figures including cubes in this clear lesson:

Exercises

1. Find the volume of a cube with \(\color{blue}{s = 6}\) cm.
2. Find the surface area of a cube with \(\color{blue}{s = 8}\) in.
3. A cube has a volume of 64 ft³. What is its side length?
4. Find the volume and surface area of a cube with \(\color{blue}{s = 2}\) m.
5. A cube has a surface area of 216 cm². Find its volume.
6. How much paint is needed to paint all 6 faces of a cube with \(\color{blue}{s = 5}\) ft?

Answers

1. \(\color{blue}{V = 6^{3}}\) = 216 cm³
2. \(\color{blue}{\text{ SA } = 6 \times 8^{2} = 6 \times 64}\) = 384 in²
3. \(\color{blue}{s^{3} = 64}\), so \(\color{blue}{s = 4}\) ft
4. \(\color{blue}{V = 2^{3}}\) = 8 m³; \(\color{blue}{\text{ SA } = 6 \times 4}\) = 24 m²
5. \(\color{blue}{6s^{2} = 216}\), \(\color{blue}{s^{2} = 36}\), \(\color{blue}{s = 6}\) cm; \(\color{blue}{V = 6^{3}}\) = 216 cm³
6. \(\color{blue}{\text{ SA } = 6 \times 5^{2}}\) = 150 ft²

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Frequently Asked Questions

What is the difference between volume and surface area?

Volume measures the three-dimensional space inside a shape (cubic units), while surface area measures the total area of all the outer faces (square units). Volume uses s³; surface area uses 6s² for a cube.

Is a cube the same as a rectangular prism?

A cube is a special type of rectangular prism where all three dimensions are equal. Every cube is a rectangular prism, but not every rectangular prism is a cube.

How do you find the side length of a cube if you know its volume?

Take the cube root of the volume: \(\color{blue}{s = ^{3}}\)√V. For example, if \(\color{blue}{V = 125}\) m³, then \(\color{blue}{s = ^{3}}\)√\(\color{blue}{125 = 5}\) m.

Related Topics

• Rectangular Prisms
• Word Problems Involving Volume of Cubes and Rectangular Prisms
• Finding Area of Compound Figures
• Area of a Parallelogram
• Polygons