Cones & spheres · Ages 14–15
Volume of a sphere
V = 43 πr³; hemisphere 23 πr³.
How to do it
Method
V = 4/3 πr³; hemisphere 2/3 πr³
r = 3: 43 × 27π = 36π cm³
Another way
2/3 of the cylinder
23 × 54π = 36π
In full
A sphere's volume = 43 πr³, r its radius. Radius 3 cm: 43 × π × 27 = 36π cm³; with π as 3.14, 113.04 cm³. A hemisphere is half of it: 23 πr³. Given the diameter, halve it first; r³ is r × r × r. Or remember Archimedes' find: a sphere fits exactly in a cylinder of radius r and height 2r, and holds 23 of it: the cylinder is πr² × 2r = 2πr³, and 23 × 2πr³ = 43 πr³. Radius 3: the cylinder is 54π, 23 × 54π = 36π, the same. The other way round, from the volume: r³ = 3V ÷ (4π), then the cube root.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The sphere in the picture has a radius of 18 cm. What is its volume, in terms of π?
Show the answer and working
Answer: 7776 π cm³
43 × π × 183 = 43 × 5832π = 7776π cm³.
Example 2
The sphere in the picture has a volume of 972π cm³. What is its radius, in cm?
Show the answer and working
Answer: 9 cm
43 × πr³ = 972π, so r³ = 3 × 972 ÷ 4 = 729 and r = 9 cm.
Example 3
A sphere of radius 12 cm has a volume of 6912π cm³.
True or false?
Show the answer and working
Answer: False
43 × π × 123 = 2304π cm³.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The hemisphere in the picture has a radius of 15 cm. What is its volume, in terms of π?
Answer:
2.The sphere in the picture is 24 cm across. What is its volume?
- A. 18432π cm³
- B. 2304π cm³
- C. 576π cm³
- D. 192π cm³
Choose the right answer
Answer:
3.The sphere in the picture has a volume of 4500π cm³. What is its radius, in cm?
Answer:
4.A cylinder holds a ball exactly: the ball touches its top, its bottom and its side. The cylinder's volume is 1458π cm³. What is the ball's volume, in terms of π?
Answer:
5.The hemisphere in the picture is 19 cm across. Correct to 1 decimal place, what is its volume, in cm³?
Answer:
All the answers
- 1. 2250 π cm³
- 2. B. 2304π cm³
- 3. 15 cm
- 4. 972 π cm³
- 5. 1795.7 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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