GCSE Maths · Higher · Chapter 9: Length, area and volume

Circles, sectors, areas, prisms and solids

Grades 4–9About 85 minutesCalculator allowed94 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Work out the area of a triangle with base 9 cm and perpendicular height 7 cm.
[1 mark]
Answer

A2
Change $3.5$ m² into cm².
[1 mark]
Answer
A3
Work out the circumference of a circle with diameter 8 cm. Give your answer correct to 1 decimal place.
[2 marks]
Answer
A4
Work out the volume of a cuboid that measures 6 cm by 4 cm by $2.5$ cm.
[1 mark]
Answer
A5
Work out the area of a circle with radius 5 cm. Give your answer in terms of $π$ .
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

A sector has radius 9 cm and angle $8 0^{\circ}$ . Work out its area. Give your answer correct to 3 significant figures.

Not drawn accurately

Hint A sector is a fraction of the whole circle. Here the fraction is

\frac{80}{360}

B1
The diagram shows a sector of a circle. Work out the length of its arc. Give your answer correct to 3 significant figures.
[2 marks]
Not drawn accurately
Answer

Example 2

A trapezium has parallel sides of 6 cm and 11 cm. The perpendicular distance between them is 5 cm. Work out its area.

Not drawn accurately

Hint Add the parallel sides, halve, and multiply by the height.

B2
Work out the area of the trapezium.
[2 marks]
Not drawn accurately
Answer

Example 3

A cylinder has radius 4 cm and height 10 cm. Work out its volume. Give your answer correct to 3 significant figures.

Not drawn accurately

Hint A cylinder is a prism. Its cross-section is a circle, so the volume is

π r^{2} h

B3
The diagram shows a closed cylinder. Work out its total surface area. Give your answer correct to 3 significant figures.
[3 marks]
Not drawn accurately
Answer

Example 4

A cone has base radius 5 cm and perpendicular height 12 cm. Work out its volume. Give your answer correct to 3 significant figures.

Not drawn accurately

Hint A cone holds one third of the cylinder with the same base and height.

B4
A sphere has radius $4.5$ cm. Work out its volume. Give your answer correct to 3 significant figures.
[2 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
A circle has an area of 50 cm². Work out its radius. Give your answer correct to 3 significant figures.
[2 marks]
Answer

C2
Work out the area of the parallelogram.
[2 marks]
Not drawn accurately
Answer
C3
The shape is a rectangle with a semicircle on one end. Work out its perimeter. Give your answer correct to 3 significant figures.
[3 marks]
Not drawn accurately
Answer
C4
The diagram shows a sector of a circle.
Not drawn accurately
(a)
Work out the arc length. Give your answer correct to 3 significant figures.
[2 marks]
Answer
(b)
Work out the perimeter of the sector.
[1 mark]
Answer
(c)
Work out the area of the sector. Give your answer correct to 3 significant figures.
[2 marks]
Answer
Total for C4: 5 marks
C5
The diagram shows a triangular prism.
Not drawn accurately
(a)
Work out the volume of the prism.
[2 marks]
Answer
(b)
Work out the total surface area of the prism.
[3 marks]
Answer
Total for C5: 5 marks
C6
A cylindrical jug has radius 5 cm. It holds exactly 1 litre of water when full. Work out the height of the jug. Give your answer correct to 3 significant figures.
[3 marks]
Answer
C7
The diagram shows a pyramid with a square base. The perpendicular height of the pyramid is 14 cm. Work out its volume.
[2 marks]
Not drawn accurately
Answer
C8
A cone has base radius 5 cm and slant height 13 cm. Work out the area of its curved surface. Give your answer correct to 3 significant figures.
[2 marks]
Answer
C9
The diagram shows a solid hemisphere. Work out its total surface area, including the flat face. Give your answer correct to 3 significant figures.
[3 marks]
Not drawn accurately
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
A trapezium has parallel sides of 7 cm and 11 cm. Its area is 72 cm². Work out the perpendicular distance between the parallel sides.
[3 marks]
Answer

D2
Mia says, "A sphere of radius 3 cm holds more than a cylinder of radius 3 cm and height 4 cm."
Show that Mia is wrong.
[3 marks]
D3
A water trough is a prism. Its cross-section is a semicircle of radius 30 cm. The trough is $1.8$ m long.
Not drawn accurately
(a)
Work out the volume of the trough in litres. Give your answer correct to 3 significant figures.
[4 marks]
Answer
(b)
Water flows into the empty trough at 12 litres per minute. How long does it take to fill the trough? Give your answer to the nearest minute.
[2 marks]
Answer
Total for D3: 6 marks
D4
A sector of a circle has radius 12 cm. Its area is $56 π$ cm². Work out the angle of the sector.
[3 marks]
Answer
D5
The solid is made from a cylinder with a cone on top. They have the same radius. The cone is 3 cm tall. Work out the volume of the solid. Give your answer correct to 3 significant figures.
[4 marks]
Not drawn accurately
Answer
D6
A solid metal sphere of radius 9 cm is melted down. All the metal is made into a cylinder of radius 6 cm. Work out the height of the cylinder.
[3 marks]
Answer
D7
The diagram shows a square of side 12 cm. A quarter circle of radius 12 cm is drawn with its centre at one corner. Work out the area of the shaded region. Give your answer correct to 3 significant figures.
[3 marks]
Not drawn accurately
Answer
D8
The diagram shows a solid cone. Work out its total surface area. Give your answer correct to 3 significant figures.
[4 marks]
Not drawn accurately
Answer
D9
A circular pond has radius $4.5$ m. A path $1.5$ m wide is laid all the way round the edge of the pond. Paving the path costs £18 per m². Work out the cost of paving the path. Give your answer to the nearest pound.
[4 marks]
Answer£ $\dots\dots$
D10
A closed cylindrical tank is to be painted on the outside, including both ends. One tin of paint covers 8 m². Work out the number of tins needed.
[4 marks]
Not drawn accurately
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
A square is drawn inside a circle of radius 5 cm. All four corners of the square are on the circle. Work out the area inside the circle but outside the square. Give your answer correct to 3 significant figures.
[3 marks]
Not drawn accurately
Hint 1 · What to try
The diagonals of the square are diameters of the circle.
Hint 2 · The first line
Each diagonal is 10 cm. A square with diagonal 10 cm can be split into two triangles with base 10 cm and height 5 cm.
Hint 3 · The full method
The square has area 50 cm² and the circle has area $25 π$ cm².
Answer

E2
A sector of radius 10 cm and angle $28 8^{\circ}$ is cut from paper. The two straight edges are joined to make a cone with no base. Work out the volume of the cone. Give your answer correct to 3 significant figures.
[4 marks]
Hint 1 · What to try
The arc of the sector becomes the circumference of the base of the cone.
Hint 2 · The first line
The arc is $\frac{288}{360} \times 2 π \times 10 = 16 π$ , so the base radius is 8 cm.
Hint 3 · The full method
The 10 cm radius becomes the slant height, so the height is $1 0^{2} - 8^{2}$ .
Answer
E3
A solid cylinder has radius $r$ and height $2 r$ . A solid sphere has the same volume as the cylinder.
Work out the radius of the sphere in terms of $r$ . Give the number correct to 3 significant figures.
[4 marks]
Hint 1 · What to try
Write the volume of the cylinder in terms of $r$ .
Hint 2 · The first line
The cylinder has volume $π r^{2} \times 2 r = 2 π r^{3}$ .
Hint 3 · The full method
Set $\frac{4}{3} π R^{3} = 2 π r^{3}$ and solve for $R$ .
Answer
E4
$A B C D$ is a square of side 6 cm. One quarter circle has its centre at $A$ and radius 6 cm. Another quarter circle has its centre at $C$ and radius 6 cm. Work out the area of the shaded region where the two quarter circles overlap. Give your answer correct to 3 significant figures.
[4 marks]
Not drawn accurately
Hint 1 · What to try
The diagonal $B D$ cuts the shaded region into two equal pieces.
Hint 2 · The first line
Each piece is a quarter circle with the triangle $A B D$ taken away: $\frac{1}{4} \times π \times 6^{2} - \frac{1}{2} \times 6 \times 6$ .
Hint 3 · The full method
That is $9 π - 18$ for each piece, so the region is $2 (9 π - 18)$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the circumference and area of a circle, and the arc length and area of a sector.
I can work out the area of a parallelogram and a trapezium.
I can work out the volume and surface area of prisms and cylinders.
I can work out the volume and surface area of pyramids, cones and spheres.

Answers and mark scheme

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