1
1
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SCALE
The diagram shows a rectangle with 6 congruent circles inside.
Each circle touches the adjacent circles and the sides of the rectangle.
The radius of each circle is 8 cm.
(a) Show that the length of the rectangle is 48 cm.
[1]
(b) Find the area of the rectangle.
Give the units of your answer.
.............................. .................... [3]
(c) Calculate the percentage of the rectangle that is shaded.
................................................... % [3]
[Total: 7]
2
POQ is a sector of a circle, centre O and radius 7.5 cm.
The sector angle is 60°.
Calculate the length of the arc PQ.
PQ = ................................................... cm [2]
[Total: 2]
3 ABCDEFGH is a regular octagon with sides of length 6 cm.
The diagram shows part of the octagon.
O is the centre of the octagon and M is the midpoint of AB.
A M B
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O
3
(a) (i) Show that angle OAM is 67.5°.
[2]
(ii) Calculate the area of the octagon.
................................................... cm2 [4]
(b) Find the area of the circle that passes through the vertices of the octagon.
................................................... cm2 [3]
[Total: 9]
4
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4m
0.45 m
The diagram shows a horizontal container for water with a uniform cross-section.
4
The cross-section is a semicircle.
The radius of the semicircle is 0.45 m and the length of the container is 4 m.
(a) Calculate the volume of the container.
................................................... m3 [2]
(b)
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0.3 m
The greatest depth of the water in the container is 0.3 m.
The diagram shows the cross-section.
Calculate the number of litres of water in the container.
Give your answer correct to the nearest integer.
................................................... litres [6]
[Total: 8]
5 A regular 12-sided polygon has side length 6 cm.
(a) Show that one interior angle of the polygon is 150°.
[1]
(b) The polygon is enclosed by a circle, centre O, so that each vertex touches the circumference of the circle.
5
(i) Show that the radius, AO, of the circle is 11.6 cm, correct to 1 decimal place.
[3]
(ii) Calculate
6
A the circumference of the circle,
................................................... cm [2]
B the perimeter of the shaded minor segment formed by the chord AB.
................................................... cm [2]
(c) The regular 12-sided polygon is the cross-section of a prism of length 2 cm.
Calculate the volume of the prism.
................................................... cm3 [3]
[Total: 11]
6 Calculate the circumference of a circle with radius 4.7 cm.
................................................... cm [2]
[Total: 2]
7
7 A circle has a radius of 18 cm.
Calculate the circumference of this circle.
................................................... cm [2]
[Total: 2]
8
O
O
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x°
A B
2.4 cm AB
The volume of a paper cone of radius 2.4 cm is 95.4 cm3.
The paper is cut along the slant height from O to AB.
The cone is opened to form a sector OAB of a circle with centre O.
Calculate the sector angle x°.
[The volume, V, of a cone with radius r and height h is .]
................................................... [6]
[Total: 6]
8
9 A circular disc has circumference 250 cm.
Calculate the radius of the disc.
................................................... cm [2]
[Total: 2]
10 The radius of a circle is 42 cm.
Work out the circumference of the circle.
Give your answer in terms of .
...................................cm [2]
[Total: 2]
11
B
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72°
O
9 cm
A
The diagram shows a sector of a circle, centre O, radius 9 cm.
The sector angle is 72°.
Calculate the area of the sector AOB.
..................................cm2 [2]
[Total: 2]
9
12
B
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72°
O
9 cm
A
The diagram shows a sector of a circle, centre O, radius 9 cm.
The sector angle is 72°.
Calculate the length of the arc AB.
...................................cm [2]
[Total: 2]
10
13
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41°
The diagram shows two sectors of a circle.
The major sector is shaded.
The area of the major sector is 74.5 cm2.
Calculate the radius of the circle.
................................................... cm [3]
[Total: 3]
11
14
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A
SCALE
B C
The diagram shows a shape made from an equilateral triangle ABC and a sector of a circle.
Points B and C lie on the circle, centre A.
The side length of the equilateral triangle is 12.4 cm.
Work out the perimeter of the shape.
................................................... cm [3]
[Total: 3]
12
15
O
60°
NOT TO
24 cm SCALE
P Q
The diagram shows a sector OPQ of a circle with centre O and radius 24 cm.
The sector angle is 60°.
A cone is made from this sector by joining OP to OQ.
NOT TO
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P
Q
Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is .]
................................................... cm3 [6]
[Total: 6]
13
16
O
53°
NOT TO
9.5 cm
A B SCALE
X Y
The diagram shows a sector OXY of a circle with centre O and radius 9.5 cm.
The sector angle is 53°.
A lies on OX, B lies on OY and OA = OB.
(a) Show that the area of the sector is 41.7 cm2 , correct to 1 decimal place.
[2]
(b) The area of triangle OAB is of the area of sector OXY.
Calculate OA.
OA = ................................................... cm [4]
[Total: 6]
14
17 The total perimeter of a semicircle is 19.02 cm.
Calculate the radius of the semicircle.
................................................... cm [3]
[Total: 3]
18
The diagram shows a circle, centre O, radius 11 cm.
C, F, G and H are points on the circumference of the circle.
The line AD touches the circle at C and is parallel to the line EG.
B is a point on AD and angle ABO = 140°.
(a) Write down the mathematical name of the straight line AD.
................................................... [1]
15
(b) (i) Find, in terms of , the circumference of the circle.
................................................... cm [2]
(ii) Work out angle FOH.
Angle FOH = ................................................... [2]
(iii) Calculate the length of the minor arc FH.
................................................... cm [2]
(c) (i) Give a reason why angle BCO is 90°.
...................................................................................................................................... [1]
(ii) Show that BC = 13.11 cm, correct to 2 decimal places.
[3]
16
(iii) Calculate BH.
BH = ................................................... cm [3]
[Total: 14]
19
The diagram shows a sector of a circle with centre O, radius 8 cm and sector angle 165°.
(a) Calculate the total perimeter of the sector.
................................................... cm [3]
17
(b) The surface area of a sphere is the same as the area of the sector.
Calculate the radius of the sphere.
[The surface area, A, of a sphere with radius r is .]
................................................... cm [4]
(c)
A cone is made from the sector by joining OA to OB.
(i) Calculate the radius, r, of the cone.
r = ................................................... cm [2]
18
(ii) Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is .]
................................................... cm3 [4]
[Total: 13]
20
The diagram shows the surface of a garden pond, made from a rectangle and two semicircles.
The rectangle measures 3 m by 1.2 m.
(a) Calculate the area of this surface.
................................................... m2 [3]
19
(b) The pond is a prism and the water in the pond has a depth of 20 cm.
Calculate the number of litres of water in the pond.
................................................... litres [3]
(c) After a rainfall, the number of litres of water in the pond is 1007.
Calculate the increase in the depth of water in the pond.
Give your answer in centimetres.
................................................... cm [3]
[Total: 9]
