KS3 Maths · Year 8 · Chapter 10: Surface area and volume of cylinders · Lesson 3

Composite shapes

Year 8About 75 minutesCalculator allowed46 marks

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Date

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Quick questions to start.

A1
Needed today
A cylinder has radius $2$ cm and height $10$ cm. Write its volume in terms of $π$ .
[1 mark]
Answer
A2
Last lesson
A cylinder has radius $3$ cm and height $4$ cm. Write its curved surface area in terms of $π$ .
[1 mark]
Answer
A3
Needed today
Write the area of a semicircle of radius $6$ cm in terms of $π$ .
[1 mark]
Answer
A4
Year 7
Work out the volume of a cuboid $5$ cm by $4$ cm by $3$ cm.
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it. Split the solid into shapes you know. Add their volumes, or take one away. Work in terms of $π$ until the last step.

Example 1

A cylinder of radius $3$ cm and height $7$ cm stands on a cylinder of diameter $12$ cm and height $4$ cm. Work out the volume of the solid, to $3$ s.f.

Hint The bottom cylinder has radius

6

cm. Add the two volumes.

B1
A cylinder of radius $2.5$ cm and height $9$ cm stands on a cylinder of diameter $14$ cm and height $3$ cm. Work out the volume of the solid, to $3$ s.f.
[2 marks]
Answer $\dots$ cm³

Example 2

A cylinder of radius $2$ cm and height $6$ cm stands on a cylinder of diameter $10$ cm and height $3$ cm. Work out the total surface area, to $3$ s.f.

Hint The small cylinder covers part of the top of the big one, and that same area is the small one's base. So: the whole big cylinder, plus the curved surface of the small one.

B2
A cylinder of radius $1.5$ cm and height $8$ cm stands on a cylinder of diameter $8$ cm and height $5$ cm. Work out the total surface area, to $3$ s.f.
[3 marks]
Answer $\dots$ cm²

Example 3

A tube is $8$ cm long. Its outer radius is $5$ cm and its inner radius is $3.5$ cm. Work out the volume of the tube, to $3$ s.f.

Hint Outer cylinder minus inner cylinder.

B3
A tube is $15$ cm long. Its outer radius is $4.5$ cm and its inner radius is $3.5$ cm. Work out the volume of the tube, to $3$ s.f.
[2 marks]
Not drawn accurately
Answer $\dots$ cm³

Example 4

A cuboid $10$ cm wide, $18$ cm long and $6$ cm high has half a cylinder on top. Work out the volume, to $3$ s.f.

Hint The half-cylinder has diameter

10

cm, so radius

5

cm, and length

18

cm.

B4
A cuboid $8$ cm wide, $25$ cm long and $5$ cm high has half a cylinder on top. Work out the volume, to $3$ s.f.
[3 marks]
Answer $\dots$ cm³

Practice

Year 8

C1
A solid is a cylinder of radius $4$ cm and height $9$ cm with a cone of the same radius and height $6$ cm on top. The volume of a cone is $\frac{1}{3} π r^{2} h$ . Work out the volume of the solid, to $3$ s.f.
[3 marks]
Answer $\dots$ cm³
C2
A $10$ cm cube has a hole of diameter $4$ cm drilled straight through it, from front to back. Work out the volume that is left, to $3$ s.f.
[3 marks]
Answer $\dots$ cm³

C3
A closed water tank is a cylinder. On the outside it is $50$ cm across and $70$ cm high. The walls, top and base are all $1$ cm thick. How many litres does it hold, to $3$ s.f.?
[3 marks]
Answer $\dots$ litres
C4
A candle is a cylinder of wax $5$ cm across and $12$ cm long. A wick $0.5$ cm across runs through its centre for the whole length. Write the ratio volume of wick : volume of wax in the form $1 : n$ .
[3 marks]
Answer $1 : \dots$

Exam-style questions

GCSE Foundation

D1
A tube is $10$ cm long. Its outer radius is $6$ cm and its inner radius is $3$ cm. Lee says, "The inner radius is half the outer radius, so the tube is half the volume of a solid cylinder of radius $6$ cm."
Show that Lee is wrong. What fraction is it?
[3 marks]

D2
Cans of soup are $8$ cm across and $12$ cm tall. They stand upright in a box $24$ cm long, $16$ cm wide and $12$ cm tall.
(a)
How many cans fit in the box?
[1 mark]
Answer $\dots$ cans
(b)
What percentage of the box do the cans fill? Give your answer to $1$ d.p.
[3 marks]
Answer $\dots$ %
Total for D2: 4 marks
D3
A garden shed is half a cylinder lying on its flat side. It has radius $1.5$ m and length $4$ m.
(a)
Work out the volume of the shed, to $3$ s.f.
[2 marks]
Answer $\dots$ m³
(b)
Work out the total surface area of the shed, including the floor, to $3$ s.f.
[3 marks]
Answer $\dots$ m²
Total for D3: 5 marks
D4
A $6$ cm cube has a hole of diameter $2$ cm drilled straight through it. Work out its total surface area, including the inside of the hole, to $3$ s.f.
[4 marks]
Answer $\dots$ cm²

Extension

Stretch

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

E1
A toy is a cylinder of radius $5$ cm and height $12$ cm with a cone on top. The toy is $20$ cm tall altogether. The curved surface of a cone is $π r l$ , where $l$ is its slant height. Work out the total surface area of the toy, to $3$ s.f.
[4 marks]
Hint 1 · What to try
The cone is $20 - 12 = 8$ cm tall. Its slant height is $5^{2} + 8^{2} = 9.43$ cm.
Hint 2 · The first line
The surface is the cone's curved surface, the cylinder's curved surface and one circle at the bottom.
Hint 3 · The full method
Cone $π \times 5 \times 9.43 = 148.2$ , cylinder $2 π \times 5 \times 12 = 377.0$ , base $π \times 5^{2} = 78.5$ . Add them.
Answer $\dots$ cm²

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can split a composite solid into cylinders, cones and cuboids.
I can find the volume of a solid by adding parts or taking one away.
I can find a surface area without counting faces that are hidden.
I can solve problems about tubes, tanks and packing.

Answers and mark scheme

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