KS3 Maths · Year 8 · Chapter 10: Surface area and volume of cylinders · Lesson 2

Surface area of a cylinder

Year 8About 70 minutesCalculator allowed45 marks

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Date

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Quick questions to start.

A1
Needed today
Work out the circumference of a circle of radius $4$ cm, to $1$ d.p.
[1 mark]
Answer
A2
Needed today
A rectangle is $12.5$ cm long and $8$ cm wide. Work out its area.
[1 mark]
Answer
A3
Last lesson
A cylinder has radius $3$ cm and height $5$ cm. Write its volume in terms of $π$ .
[1 mark]
Answer
A4
Year 7
How many cm² are there in $1$ m²?
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it. Opened out, the curved surface is a rectangle $2 π r$ long and $h$ wide, so its area is $2 π r h$ . The total surface area adds the two circles: $2 π r h + 2 π r^{2}$ .

Example 1

Work out the curved surface area of this cylinder, to $3$ s.f.

Hint The rectangle is as long as the circumference,

2 π r

, and as wide as the height.

B1
Work out the curved surface area of this cylinder, to $3$ s.f.
[2 marks]
Answer $\dots$ cm²

Example 2

Work out the total surface area of this cylinder, to $3$ s.f.

Hint Curved surface

2 π r h

, then add two circles,

2 π r^{2}

B2
Work out the total surface area of this cylinder, to $3$ s.f.
[2 marks]
Answer $\dots$ m²

Example 3

Work out the total surface area of this cylinder. Give your answer in terms of $π$ .

Hint The diameter is

8

cm, so the radius is

4

cm.

B3
Work out the total surface area of this cylinder. Give your answer in terms of $π$ .
[3 marks]
Answer $\dots$ cm²

Example 4

The curved surface area of this cylinder is $250$ cm². Work out its height $h$ , to $1$ d.p.

Not drawn accurately

Hint

2 π r h = 250

. Work out

2 π r

first, then divide.

B4
The curved surface area of this cylinder is $500$ cm². Work out its height $h$ , to $1$ d.p.
[3 marks]
Not drawn accurately
Answer $h = \dots$ cm

Practice

Year 8

C1
A tin has a radius of $3.6$ cm and a height of $10.5$ cm. A paper label covers the curved surface, with an extra $1.5$ cm overlap for the glue. Work out the area of the label, to the nearest cm².
[3 marks]
Answer $\dots$ cm²
C2
A plant pot is a cylinder with no top. Its diameter is $18$ cm and its height is $16$ cm. Work out the area of its outside surface, to $3$ s.f.
[3 marks]
Answer $\dots$ cm²

C3
A cone has radius $7$ cm and height $24$ cm. Its total surface area is $π r l + π r^{2}$ , where $l$ is the slant height. Work out the total surface area, to $3$ s.f.
[3 marks]
Answer $\dots$ cm²
C4
A paper straw is a cylinder, open at both ends. Its diameter is $6$ mm and its length is $21$ cm. Work out the area of paper in it, in cm², to $3$ s.f.
[2 marks]
Answer $\dots$ cm²

Exam-style questions

GCSE Foundation

D1
Mia says, "Cylinders $A$ and $B$ use the same two numbers, so they have the same total surface area."
Show that Mia is wrong.
[3 marks]

D2
A closed oil tank is a cylinder of radius $1.2$ m and height $3.5$ m. Its whole outside, top and bottom too, is painted. One tin of paint covers $6$ m². How many tins are needed?
[4 marks]
Answer $\dots$ tins
D3
A pipe is $8$ m long and has an outside diameter of $30$ cm. The curved outside surface is to be painted.
Not drawn accurately
(a)
Work out the area to be painted, in m², to $2$ d.p.
[3 marks]
Answer $\dots$ m²
(b)
$1$ litre of paint covers $3$ m². The paint is sold in $1$ litre tins. How many tins are needed?
[2 marks]
Answer $\dots$ tins
Total for D3: 5 marks
D4
A closed cylinder has a radius of $5$ cm and a total surface area of $400$ cm². Work out its height $h$ , to $3$ s.f.
[4 marks]
Not drawn accurately
Answer $h = \dots$ cm

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A sheet of card $24$ cm by $15$ cm is rolled into an open tube, with no overlap. It can be rolled so that the tube is $15$ cm tall, or so that it is $24$ cm tall. Which tube holds more, and by how much? Give your answer to $3$ s.f.
[4 marks]
Hint 1 · What to try
The edge that goes round becomes the circumference. For the $15$ cm tall tube, $2 π r = 24$ .
Hint 2 · The first line
$15$ cm tall: $r = 24 \div 2 π = 3.820$ and $V = π \times 3.82 0^{2} \times 15 = 687.5$ cm³.
Hint 3 · The full method
$24$ cm tall: $r = 15 \div 2 π = 2.387$ and $V = π \times 2.38 7^{2} \times 24 = 429.7$ cm³. The difference is $257.8$ .
Answer $\dots$ cm³

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the curved surface area of a cylinder with $2 π r h$ .
I can find the total surface area of a closed or an open cylinder.
I can work back from a surface area to the height.
I can use surface areas for labels, paint and nets.

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