Prisms · Ages 12–13
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
A prism has a base area of 11 cm², and its two ends are 11 cm apart. What is its volume, in cm³?
Show the answer and working
Answer: 121 cm³
A prism's volume = base area × height (the distance between its ends) = 11 × 11 = 121 cm³.
Example 2
The prism in the picture has a right-angled triangle as its cross-section, its two shorter sides 9 cm and 8 cm, and it is 15 cm long. What is its volume, in cm³?
Show the answer and working
Answer: 540 cm³
Cross-section 9 × 8 ÷ 2 = 36 cm², volume = cross-section × length = 36 × 15 = 540 cm³.
Example 3
A prism 6 cm long whose cross-section is a triangle 4 cm across and 5 cm high has a volume of 60 cm³.
True or false?
Show the answer and working
Answer: True
The cross-section is 4 × 5 ÷ 2, times the length 6: 60 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 triangle 8 cm across and 9 cm high as its cross-section, and is 19 cm long. What is its volume?
- A. 8 × 9 ÷ 2 × 19 = 684 cm³
- B. 8 × 9 × 19 = 1368 cm³
- C. 8 × 9 ÷ 2 + 19 = 55 cm³
- D. (8 + 9) × 19 = 323 cm³
Choose the right answer
Answer:
2.The prism in the picture has a right-angled triangle as its cross-section, its two shorter sides 12 cm and 7 cm, and it is 15 cm long. What is its volume, in cm³?
Answer:
3.The prism in the picture has a trapezium as its cross-section: 5 cm along the top, 9 cm along the bottom, 7 cm high. It is 6 cm long. What is its volume, in cm³?
Answer:
4.A prism 5 cm long whose cross-section is a triangle 9 cm across and 6 cm high has a volume of 270 cm³.
True or false?
Answer:
5.The prism in the picture is 6 cm long with a volume of 480 cm³, its cross-section a trapezium 6 cm along the top and 14 cm along the bottom. How high is the trapezium, in cm?
Answer:
All the answers
- 1. A. 8 × 9 ÷ 2 × 19 = 684 cm³
- 2. 630 cm³
- 3. 294 cm³
- 4. False
- 5. 8 cm
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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