Cones & spheres · Ages 14–15
Volume of a pyramid
V = 13 × base area × height.
How to do it
Method
V = 1/3 × base area × height
13 × 6 × 5 × 8 = 80 cm³
Another way
The prism, then ÷ 3
30 × 8 = 240 → 240 ÷ 3 = 80
In full
A pyramid narrows from its base all the way to a point. A prism on the same base and as high holds 3 pyramids of water, so a pyramid's volume = 13 × base area × height. The height is the upright distance from the top to the base, not a slanting edge. A pyramid on a 6 cm × 5 cm rectangle, 8 cm high: the base is 30 cm², the volume 13 × 30 × 8 = 80 cm³. Or work out the prism on the same base and as high, then divide by 3: 30 × 8 = 240, 240 ÷ 3 = 80 cm³. A triangular pyramid's base is a triangle: base × height ÷ 2 first. The other way round, from the volume: height = 3 × volume ÷ base area.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A pyramid has a base area of 90 cm² and a height of 24 cm. What is its volume, in cm³?
Show the answer and working
Answer: 720 cm³
13 × base area × height = 13 × 90 × 24 = 720 cm³.
Example 2
A triangular pyramid stands on a triangle 10 cm across and 10 cm high, and the pyramid is 18 cm high. What is its volume, in cm³?
Show the answer and working
Answer: 300 cm³
The base is 10 × 10 ÷ 2 = 50 cm², the volume 13 × 50 × 18 = 300 cm³.
Example 3
A pyramid on a 7 cm × 6 cm rectangle, 17 cm high, has a volume of 238 cm³.
True or false?
Show the answer and working
Answer: True
13 × 7 × 6 × 17 = 238 cm³; 7 × 6 × 17 is the prism on the same base.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The pyramid in the picture stands on a square of side 11 cm and is 18 cm high. What is its volume?
- A. 264 cm³
- B. 726 cm³
- C. 2178 cm³
- D. 1089 cm³
Choose the right answer
Answer:
2.A pyramid has a base area of 69 cm² and a height of 10 cm. What is its volume, in cm³?
Answer:
3.The pyramid in the picture stands on a square of side 12 cm and has a volume of 672 cm³. How high is it, in cm?
Answer:
4.A pyramid on a 11 cm × 3 cm rectangle, 12 cm high, has a volume of 396 cm³.
True or false?
Answer:
5.The pyramid in the picture stands on a square of side 72 cm, and each side face's slant height (from the top to the middle of a base edge) is 39 cm. What is its volume, in cm³?
Answer:
All the answers
- 1. B. 726 cm³
- 2. 230 cm³
- 3. 14 cm
- 4. False
- 5. 25920 cm³
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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