How to Find the Volume and Surface Area of Rectangular Prisms? (+FREE Worksheet!)
A rectangular prism is the 3-D shape you encounter every day — think of a cereal box, a brick, or a room. It has six rectangular faces, three pairs of parallel and congruent faces, and all right-angle corners. On the GED Math test you need to know how to find its volume (interior space) and its surface area (total face area). Both formulas are straightforward once you know the three dimensions.
What Is a Rectangular Prism?
A rectangular prism (also called a cuboid) is a solid with:
- 6 rectangular faces
- 12 edges
- 8 vertices (corners)
- Three dimensions: length (l), width (w), and height (h)
When \(\color{blue}{l = w = h}\), the rectangular prism becomes a cube.
Formulas for a Rectangular Prism
Volume of a Rectangular Prism
\(\color{blue}{V = l \times w \times h}\)
Volume is measured in cubic units (cm³, in³, ft³, etc.).
Surface Area of a Rectangular Prism
\(\color{blue}{\text{ SA } = 2(\text{ lw } + \text{ lh } + \text{ wh })}\)
Surface area is measured in square units (cm², in², ft², etc.).
The formula adds the areas of all 6 faces: \(\color{blue}{\text{ top } + \text{ bottom }}\) (lw each), \(\color{blue}{\text{ front } + \text{ back }}\) (lh each), and two sides (wh each).
Step-by-Step Summary
- Identify the length, width, and height from the problem or diagram.
- For volume: multiply \(\color{blue}{l \times w \times h}\).
- For surface area: compute lw, lh, and wh; add them; multiply by 2.
- Label the answer with the correct units (cubic for volume, square for SA).
Watch: Volume of Rectangular Prisms (Video Lesson)
Math with Mr. J explains the volume formula with step-by-step worked examples:
Worked Examples
Example 1: A rectangular prism has \(\color{blue}{l = 6}\) cm, \(\color{blue}{w = 4}\) cm, \(\color{blue}{h = 3}\) cm. Find the volume and surface area.
\(\color{blue}{V = 6 \times 4 \times 3}\) = 72 cm³
\(\color{blue}{\text{ SA } = 2(6\times 4 + 6\times 3 + 4\times 3) = 2(24 + 18 + 12) = 2(54)}\) = 108 cm²
Example 2: A box measures 8 \(\color{blue}{\text{ in } \times 5}\) \(\color{blue}{\text{ in } \times 2}\) in. Find its volume and surface area.
\(\color{blue}{V = 8 \times 5 \times 2}\) = 80 in³
\(\color{blue}{\text{ SA } = 2(8\times 5 + 8\times 2 + 5\times 2) = 2(40 + 16 + 10) = 2(66)}\) = 132 in²
Example 3: Find the volume of a prism with \(\color{blue}{l = 10}\) m, \(\color{blue}{w = 3}\) m, \(\color{blue}{h = 4}\) m.
\(\color{blue}{V = 10 \times 3 \times 4}\) = 120 m³
Example 4: A fish tank is 7 \(\color{blue}{\text{ ft } \times 2}\) \(\color{blue}{\text{ ft } \times 5}\) ft. How many cubic feet of water does it hold?
\(\color{blue}{V = 7 \times 2 \times 5}\) = 70 ft³
More Practice: Surface Area for GED Math (Video)
This GED-focused lesson walks you through finding the surface area of a rectangular prism:
Exercises
- A rectangular prism has \(\color{blue}{l = 5}\) ft, \(\color{blue}{w = 3}\) ft, \(\color{blue}{h = 2}\) ft. Find the volume.
- Find the surface area of a prism with \(\color{blue}{l = 9}\) cm, \(\color{blue}{w = 4}\) cm, \(\color{blue}{h = 6}\) cm.
- A box has \(\color{blue}{V = 120}\) in³, \(\color{blue}{l = 10}\) in, \(\color{blue}{w = 4}\) in. Find h.
- Find the volume of a prism with \(\color{blue}{l = 12}\) m, \(\color{blue}{w = 5}\) m, \(\color{blue}{h = 3}\) m.
- Find the surface area of a prism with \(\color{blue}{l = 7}\) in, \(\color{blue}{w = 2}\) in, \(\color{blue}{h = 5}\) in.
- A shipping container is 40 \(\color{blue}{\text{ ft } \times 8}\) \(\color{blue}{\text{ ft } \times 8}\) ft. What is its volume?
Answers
- \(\color{blue}{V = 5 \times 3 \times 2}\) = 30 ft³
- \(\color{blue}{\text{ SA } = 2(9\times 4 + 9\times 6 + 4\times 6) = 2(36 + 54 + 24) = 2(114)}\) = 228 cm²
- \(\color{blue}{h = 120 \div (10 \times 4) = 120 \div 40}\) = 3 in
- \(\color{blue}{V = 12 \times 5 \times 3}\) = 180 m³
- \(\color{blue}{\text{ SA } = 2(7\times 2 + 7\times 5 + 2\times 5) = 2(14 + 35 + 10) = 2(59)}\) = 118 in²
- \(\color{blue}{V = 40 \times 8 \times 8}\) = 2,560 ft³
Frequently Asked Questions
What is the difference between a rectangular prism and a cube?
A cube is a special rectangular prism where all three dimensions (length, width, height) are equal. An ordinary rectangular prism can have different values for l, w, and h.
How do I remember the surface area formula?
Think of the three unique pairs of faces: top/bottom (lw), front/back (lh), and left/right (wh). Add those three products, then double the sum since each pair has two identical faces: \(\color{blue}{\text{ SA } = 2(\text{ lw } + \text{ lh } + \text{ wh })}\).
What units do I use for volume vs. surface area?
Volume uses cubic units (in³, cm³, ft³). Surface area uses square units (in², cm², ft²). Always check the unit of the side lengths and cube or square it accordingly.
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