Volume of Prisms
A prism is a 3D shape with a constant cross-section. Understanding how to calculate volume is essential for countless real-world tasks — from working out how much water fits in a swimming pool to calculating the amount of concrete needed for a building.
The General Formula for Any Prism
The key to finding the volume of any prism is to first find the area of its cross-section — the identical shape at each end — then multiply by the length.
Think of a prism like a sliced loaf of bread — every slice (the cross-section) is identical. Whatever that shape is, find its area, then multiply by how long the loaf is.
Key Prism Formulas
Cuboid
Cross-section is a rectangle ($l \times w$).
Cylinder
Cross-section is a circle ($\pi r^2$).
Remember: use the radius, not the diameter!
Triangular Prism
Cross-section is a triangle ($\frac{1}{2} \times b \times h$).
Here $h_{\triangle}$ is the triangle’s perpendicular height, and $l$ is the prism’s length.
Worked Examples
Example 1: Volume of a Cylinder
A cylinder has a radius of 1.5 m and a height of 2.5 m. Calculate its volume to 3 s.f.
- Formula: $V = \pi r^2 h$.
- Substitute: $V = \pi \times (1.5)^2 \times 2.5$.
- Calculate: $V = \pi \times 2.25 \times 2.5 = 17.671\ldots$
Answer: $17.7 \text{ m}^3$ (to 3 s.f.)
Example 2: Volume of a Triangular Prism
A tent has a triangular cross-section with base 2.4 m and height 1.8 m. The tent is 3 m long. Find its volume.
- Cross-section area: $\frac{1}{2} \times 2.4 \times 1.8 = 2.16 \text{ m}^2$.
- Multiply by length: $V = 2.16 \times 3 = 6.48$.
Answer: $6.48 \text{ m}^3$
Example 3: Finding a Missing Dimension
A swimming pool shaped like a cuboid has a volume of 120 m³. Its length is 10 m and width is 4 m. Find its depth.
- Formula: $V = l \times w \times h$.
- Substitute known values: $120 = 10 \times 4 \times h$.
- Simplify: $120 = 40h$.
- Solve: $h = 120 \div 40 = 3$.
Answer: The depth is 3 m.
Tutor Insights
🤔 Common Misunderstandings
- Confusing area and volume: Students stop after finding the cross-section area and forget to multiply by the length — or use square units (cm²) for a volume answer.
- Identifying the cross-section: Struggling to spot which face is the cross-section, especially when a prism is lying on its side. The cross-section is the identical face repeated from one end to the other.
📝 Common Exam Mistakes
- Using the diameter instead of the radius for cylinders. Always halve the diameter if the question gives you the full width.
- Incorrect units in the final answer — volume is cubic (cm³, m³), not squared.
- Rounding too early — only round your final answer to the precision stated in the question.
Practice Questions
- A gift box is a cuboid measuring 15 cm by 10 cm by 8 cm. Calculate its volume.
- A cylindrical plant pot has a radius of 12 cm and a height of 20 cm. Calculate its volume to 1 decimal place.
- A fish tank in the shape of a cuboid has a volume of $18{,}000 \text{ cm}^3$. Its length is 60 cm and its height is 30 cm. Calculate its width.
Show Answers
- Working: $V = 15 \times 10 \times 8 = 1{,}200$.
Answer: 1,200 cm³. - Working: $V = \pi \times 12^2 \times 20 = 2880\pi \approx 9{,}047.8$.
Answer: 9,047.8 cm³. - Working: $18{,}000 = 60 \times w \times 30 \implies 18{,}000 = 1{,}800w \implies w = 10$.
Answer: 10 cm.
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