Prisms & cylinders · Ages 13–14
Volume of a prism
Volume = cross-section area × length.
How to do it
Method
Volume = cross-section area × length
6 × 8 ÷ 2 = 24, 24 × 10 = 240 cm³
Another way
Two make a cuboid
6 × 8 × 10 ÷ 2 = 240
In full
Cut a prism anywhere along its length and the cross-section is the same, so its volume = base area × height. Lying down, its 'height' is its length: the distance between its two ends. Find the cross-section first (the face that matches both ends), work out its area with the flat shape's rule — a triangle base × height ÷ 2, a trapezium (top + bottom) × height ÷ 2, an L split into two rectangles — and multiply by the length. A cross-section that is a right-angled triangle with legs 6 cm and 8 cm, 10 cm long: base area 6 × 8 ÷ 2 = 24 cm², volume 24 × 10 = 240 cm³. Or think of fitting two together: two such prisms make a 6 × 8 × 10 cuboid, 480 cm³, so one is half, 240 cm³. The other way round, from the volume: length = volume ÷ base area.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The prism in the picture has a triangle 12 cm across and 4 cm high as its cross-section, and is 13 cm long. What is its volume?
- A. 12 × 4 × 13 = 624 cm³
- B. 12 × 4 ÷ 2 × 13 = 312 cm³
- C. 12 × 4 ÷ 2 + 13 = 37 cm³
- D. (12 + 4) × 13 = 208 cm³
Choose the right answer
Show the answer and working
Answer: B. 12 × 4 ÷ 2 × 13 = 312 cm³
The cross-section is 12 × 4 ÷ 2, times the length 13: 312 cm³.
Example 2
The prism in the picture has a volume of 462 cm³, and its cross-section is a triangle 6 cm across and 7 cm high. How long is the prism?
Show the answer and working
Answer: 22 cm
The cross-section is 6 × 7 ÷ 2 = 21 cm², so the length = volume ÷ cross-section = 462 ÷ 21 = 22 cm.
Example 3
The prism in the picture has a regular hexagon as its cross-section; joined from its centre to its six corners, it splits into 6 equal triangles of 29 cm² each. The cross-section is cm²; the prism is 5 cm long, so its volume is cm³.
Fill in each blank
Show the answer and working
Answer: 174, 870
Cross-section 6 × 29 = 174 cm², volume 174 × 5 = 870 cm³.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The prism in the picture has a right-angled triangle as its cross-section, its two shorter sides 12 cm and 3 cm, and it is 19 cm long. What is its volume, in cm³?
Answer:
2.The prism in the picture has a triangle 10 cm across and 7 cm high as its cross-section, and is 7 cm long. What is its volume?
- A. 10 × 7 ÷ 2 + 7 = 42 cm³
- B. (10 + 7) × 7 = 119 cm³
- C. 10 × 7 ÷ 2 × 7 = 245 cm³
- D. 10 × 7 × 7 = 490 cm³
Choose the right answer
Answer:
3.The prism in the picture has a volume of 378 cm³, and its cross-section is a triangle 12 cm across and 9 cm high. How long is the prism?
Answer:
4.The prism in the picture has a trapezium as its cross-section: 5 cm along the top, 13 cm along the bottom, 4 cm high. It is 10 cm long. What is its volume, in cm³?
Answer:
5.The prism in the picture has a regular hexagon as its cross-section; joined from its centre to its six corners, it splits into 6 equal triangles of 10 cm² each. The cross-section is cm²; the prism is 18 cm long, so its volume is cm³.
Fill in each blank
Answer:
All the answers
- 1. 342 cm³
- 2. C. 10 × 7 ÷ 2 × 7 = 245 cm³
- 3. 7 cm
- 4. 360 cm³
- 5. 60, 1080
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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