Cones & spheres · Ages 14–15
Volume of a cone
V = 13 πr²h.
How to do it
Method
V = 1/3 πr²h
r = 3, h = 10: 13 × 9π × 10 = 30π cm³
Another way
A third of the cylinder
90π ÷ 3 = 30π
In full
A cone is a pyramid on a circle: volume = 13 × base area × height = 13 πr²h. Radius 3 cm, height 10 cm: the base is 9π, the volume 13 × 9π × 10 = 30π cm³; with π as 3.14, 94.2 cm³. Or compare it with the cylinder on the same base and as high: the cylinder is 9π × 10 = 90π, the cone a third of it, 30π. Careful: given the slant height l, find the height first by Pythagoras' theorem, h² = l² − r²: radius 6, slant 10, h = 100 − 36 = 8. The other way round: h = 3V ÷ (πr²); r² = 3V ÷ (πh).
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Taking π as 3.14, what is the volume of the cone in the picture (radius 3 cm, height 18 cm), in cm³?
Show the answer and working
Answer: 169.56 cm³
13 × π × 32 × 18 = 54π; with 3.14, 169.56 cm³.
Example 2
A cone of base radius 3 cm and height 15 cm has a volume of 45π cm³.
True or false?
Show the answer and working
Answer: True
13 × π × 32 × 15 = 45π cm³: a cone is a third of the cylinder on its base.
Example 3
The cone in the picture is 12 cm high and has a volume of 196π cm³. What is its base radius, in cm?
Show the answer and working
Answer: 7 cm
13 × πr² × 12 = 196π, so r² = 3 × 196 ÷ 12 = 49 and r = 7 cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The cone in the picture is 6 cm across its base and 6 cm high. What is its volume?
- A. 6π cm³
- B. 72π cm³
- C. 18π cm³
- D. 54π cm³
Choose the right answer
Answer:
2.The cone in the picture has a base radius of 6 cm and a volume of 192π cm³. How high is it, in cm?
Answer:
3.The cone in the picture has a base radius of 9 cm and a slant height of 15 cm. What is its volume, in terms of π?
Answer:
4.The cone in the picture has a base radius of 15 cm and a height of 14 cm. Correct to 1 decimal place, what is its volume, in cm³?
Answer:
5.The cone in the picture is 8 cm high and has a volume of 24π cm³. What is its base radius, in cm?
Answer:
All the answers
- 1. C. 18π cm³
- 2. 16 cm
- 3. 324 π cm³
- 4. 3298.7 cm³
- 5. 3 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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