10. Volume of Boxes and Cubes

Volume tells how much a solid holds or fills.

Formulas:

• Rectangular prism: \(V=lwh\)
• Cube: \(V=s^3\)

Example: a box is \(5\) by \(3\) by \(2\). Volume \(=5 \times 3 \times 2=30\) cubic units.

Common mistake: adding dimensions. \(5+3+2=10\) is not volume.

Deepen your understanding
Use this focused guide to turn Volume of Boxes and Cubes into a dependable GED® problem-solving skill.

Learning targets

• Model volume as layers of unit cubes.
• Calculate rectangular-prism and cube volume with cubic units.
• Find a missing dimension when volume and two dimensions are known.

Step-by-step solving routine

1. Confirm that length, width, and height use the same unit.
2. Multiply the base rows and columns to find cubes per layer, then multiply by the number of layers.
3. For a missing height, rearrange to (h=V\div(lw)).
4. Check that changing one dimension changes volume by the same scale factor.

GED® application
A storage bin measuring (6\) ft by (4\) ft by (3\) ft holds (72\text{ ft}^3). If its length doubles while width and height stay fixed, its capacity doubles to (144\text{ ft}^3).

Coach check
Name the three multiplied dimensions before calculating; using only two dimensions produces area, not volume.

Visual study gallery
Study each diagram before the worked examples. Name what is measured, identify the needed dimensions or relationships, and explain which formula or rule the picture supports.

10 worked examples and problem solving
Use these examples as guided practice before attempting the session MCQs. Each problem is self-contained, including any needed table or context.

Example 1
Problem: A rectangular box is \(5\) by \(3\) by \(2\). What is its volume?
Answer: \(30\) cubic units
Solution: Box volume is length x width x height: \(5 \times 3 \times 2 = 30\). Pro tip: volume multiplies all three dimensions.

Example 2
Problem: A cube has side length \(4\). What is its volume?
Answer: \(64\) cubic units
Solution: Cube volume is side cubed: \(4^3 = 64\). Pro tip: surface area is 6s^\(2\), but volume is \(s^3\).

Example 3
Problem: A rectangular box is \(7\) by \(6\) by \(3\). What is its volume?
Answer: \(126\) cubic units
Solution: Box volume is length x width x height: \(7 \times 6 \times 3 = 126\). Pro tip: volume multiplies all three dimensions.

Example 4
Problem: A cube has side length \(5\). What is its volume?
Answer: \(125\) cubic units
Solution: Cube volume is side cubed: \(5^3 = 125\). Pro tip: surface area is 6s^\(2\), but volume is \(s^3\).

Example 5
Problem: A rectangular box is \(8\) by \(4\) by \(6\). What is its volume?
Answer: \(192\) cubic units
Solution: Box volume is length x width x height: \(8 \times 4 \times 6 = 192\). Pro tip: volume multiplies all three dimensions.

Example 6
Problem: A cube has side length \(6\). What is its volume?
Answer: \(216\) cubic units
Solution: Cube volume is side cubed: \(6^3 = 216\). Pro tip: surface area is 6s^\(2\), but volume is \(s^3\).

Example 7
Problem: A rectangular box is \(10\) by \(9\) by \(2\). What is its volume?
Answer: \(180\) cubic units
Solution: Box volume is length x width x height: \(10 \times 9 \times 2 = 180\). Pro tip: volume multiplies all three dimensions.

Example 8
Problem: A cube has side length \(7\). What is its volume?
Answer: \(343\) cubic units
Solution: Cube volume is side cubed: \(7^3 = 343\). Pro tip: surface area is 6s^\(2\), but volume is \(s^3\).

Example 9
Problem: A rectangular box is \(12\) by \(5\) by \(4\). What is its volume?
Answer: \(240\) cubic units
Solution: Box volume is length x width x height: \(12 \times 5 \times 4 = 240\). Pro tip: volume multiplies all three dimensions.

Example 10
Problem: A cube has side length \(8\). What is its volume?
Answer: \(512\) cubic units
Solution: Cube volume is side cubed: \(8^3 = 512\). Pro tip: surface area is 6s^\(2\), but volume is \(s^3\).

Additional resources

• Formula card: box volume \(V=lwh\), cube volume \(V=s^3\).
• Dimension check: volume should multiply three lengths and end in cubic units.