Cones & spheres · Ages 14–15
Surface area of a sphere
4πr²; a solid hemisphere 3πr².
How to do it
Method
4πr²; a solid hemisphere 3πr²
r = 5: 4π × 25 = 100π cm²
Another way
The cylinder's curved side
2π × 5 × 10 = 100π
In full
A sphere's surface area = 4πr²: exactly 4 circles of its radius. Radius 5 cm: 4π × 25 = 100π cm². Careful with a hemisphere: its curved part is half the sphere's, 2πr²; a solid hemisphere also has a flat circle πr², so its total surface area = 2πr² + πr² = 3πr². A solid hemisphere of radius 5: 50π + 25π = 75π cm², not 50π. Or remember Archimedes' cylinder again: the sphere's surface is as big as the curved side of the cylinder round it, 2πr × 2r = 4πr². The other way round: r² = surface area ÷ (4π), then the square root.
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 surface area of the sphere in the picture (radius 20 cm), in cm²?
Show the answer and working
Answer: 5024 cm²
4 × 3.14 × 202 = 5024 cm².
Example 2
The sphere in the picture has a radius of 10 cm. What is its surface area, in terms of π?
Show the answer and working
Answer: 400 π cm²
4πr² = 4 × π × 102 = 400π cm²: 4 circles of radius 10.
Example 3
A solid hemisphere of radius 17 cm has a total surface area of 867π cm².
True or false?
Show the answer and working
Answer: True
Curved 2πr² plus flat πr²: 3π × 172 = 867π cm².
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Taking π as 3.14, what is the surface area of the sphere in the picture (radius 6 cm), in cm²?
Answer:
2.The solid hemisphere in the picture has a radius of 6 cm. What is its total surface area?
- A. 72π cm²
- B. 144π cm²
- C. 108π cm²
- D. 36π cm²
Choose the right answer
Answer:
3.The sphere in the picture has a surface area of 144π cm². What is its radius, in cm?
Answer:
4.The sphere in the picture is 8 cm across. What is its surface area, in terms of π?
Answer:
5.A hollow hemispherical bowl has an outer radius of 17 cm and an inner radius of 7 cm, its rim a flat ring. What is its total surface area (outside, inside and rim), in terms of π?
Answer:
All the answers
- 1. 452.16 cm²
- 2. C. 108π cm²
- 3. 6 cm
- 4. 64 π cm²
- 5. 916 π cm²
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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