A rectangle of paper 4π cm long and 11 cm wide is rolled up along its length into a cylindrical can's side, edge to edge, and given a base. In terms of π, what is the can's capacity, in cm³?

44 π cm³. The paper's length becomes the base's circumference: 2πr = 4π, so r = 2 cm; its width is the height, 11 cm. Capacity π × 2^2 × 11 = 44π cm³.

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
Cut the curved surface of the cylinder in the picture and lay it flat: a rectangle 16 cm wide. How long is the rectangle?
- A. 10π cm
- B. 20π cm
- C. 20 cm
- D. 100π cm
Choose the right answer
Show the answer and working
Answer: B. 20π cm
Rolled up, the rectangle goes once round the base: the circumference, 2π × 10 = 20π cm.
Example 2
Using your calculator's π key, what is the total surface area of the cylinder in the picture, radius 9 cm and height 12 cm, in cm², to 3 significant figures?
Show the answer and working
Answer: 1190 cm²
In π: 2π × 9² + 2π × 9 × 12 = 378π; press π at the end: 378π ≈ 1190 cm² (3 significant figures).
Example 3
A rectangle of paper 4π cm long and 11 cm wide is rolled up along its length into a cylindrical can's side, edge to edge, and given a base. In terms of π, what is the can's capacity, in cm³?
Show the answer and working
Answer: 44 π cm³
The paper's length becomes the base's circumference: 2πr = 4π, so r = 2 cm; its width is the height, 11 cm. Capacity π × 2² × 11 = 44π cm³.

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 17 cm wide. How long is the rectangle?
- A. 12 cm
- B. 6π cm
- C. 36π cm
- D. 12π cm
Choose the right answer
Answer:
2.A cylinder of base radius 7 cm and height 11 cm has a total surface area of 252π cm².
True or false?
Answer:
3.The cylinder in the picture has a base radius of 4 cm and a curved surface area of 40π cm². How high is it, in cm?
Answer:
4.A cylinder of base radius 4 cm and height 4 cm is glued centred on top of a cylinder of base radius 6 cm and height 2 cm, making one solid. In terms of π, what is the solid's total surface area, in cm²?
Answer:
5.A rectangle of paper 10π cm long and 20 cm wide is rolled up along its length into a cylindrical can's side, edge to edge, and given a base. In terms of π, what is the can's capacity, in cm³?
Answer:

All the answers

1. D. 12π cm
2. True
3. 5 cm
4. 128 π cm²
5. 500 π cm³

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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