A cylinder has a base radius of 5 cm and a total surface area of 150π cm². How high is it, in cm?

10 cm. The ends are 2π × 5^2 = 50π, so the curved surface is 150π − 50π = 100π; height = 100π ÷ (2π × 5) = 100 ÷ 10 = 10 cm.

Prisms & cylinders · Ages 13–14

Surface area of a cylinder

Curved 2πrh; total 2πr² + 2πrh.

How to do it

Method

Curved 2πrh; total 2πr² + 2πrh

r = 3, h = 10: 60π + 18π = 78π cm²

Another way

Take out the factor

2πr(r + h) = 6π × 13 = 78π

In full

Cut a cylinder's curved surface along a straight line and lay it flat: a rectangle, as long as the base's circumference 2πr and as wide as the height h. So the curved surface area = 2πrh. For the total, add the two circular ends: 2πr² + 2πrh. Radius 3 cm, height 10 cm: curved 2π × 3 × 10 = 60π cm², the two ends 2 × π × 3² = 18π cm², 78π cm² in all. Or take out the common factor first: 2πr(r + h) = 2π × 3 × (3 + 10) = 6π × 13 = 78π cm², the same. Read which part is asked: a pipe is its curved surface only; a cup with no lid one end and the curved surface, πr² + 2πrh; halve a diameter first.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
The cylinder in the picture has a base radius of 10 cm and a height of 9 cm. What is its total surface area, in terms of π?
Show the answer and working
Answer: 380 π cm²
Curved 2π × 10 × 9 = 180π, the two ends 2 × π × 10² = 200π, 380π cm² in all.
Example 2
Cut the curved surface of the cylinder in the picture and lay it flat: a rectangle 4 cm wide. How long is the rectangle?
- A. 18 cm
- B. 9π cm
- C. 18π cm
- D. 81π cm
Choose the right answer
Show the answer and working
Answer: C. 18π cm
Rolled up, the rectangle goes once round the base: the circumference, 2π × 9 = 18π cm.
Example 3
A cylinder has a base radius of 5 cm and a total surface area of 150π cm². How high is it, in cm?
Show the answer and working
Answer: 10 cm
The ends are 2π × 5² = 50π, so the curved surface is 150π − 50π = 100π; height = 100π ÷ (2π × 5) = 100 ÷ 10 = 10 cm.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Cut the curved surface of the cylinder in the picture and lay it flat: a rectangle 9 cm wide. How long is the rectangle?
- A. 24π cm
- B. 24 cm
- C. 144π cm
- D. 12π cm
Choose the right answer
Answer:
2.The cylinder in the picture has a base radius of 2 cm and a height of 17 cm. What is its curved surface area, in terms of π?
Answer:
3.The cylinder in the picture has a base radius of 7 cm and a height of 3 cm. What is its total surface area, in terms of π?
Answer:
4.The picture shows a cylindrical cup with no lid, base radius 4 cm, height 5 cm. What is the area of its outside (the base and the side), in terms of π?
Answer:
5.The cylinder in the picture has a base radius of 9 cm and a curved surface area of 216π cm². How high is it, in cm?
Answer:

All the answers

1. A. 24π cm
2. 68 π cm²
3. 140 π cm²
4. 56 π cm²
5. 12 cm

Free worksheet: Surface area of a cylinderWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

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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