14
1
8 [Volume of cone = rr 2 h ]
3
[Curved surface area of a cone = rrl ]
l h
The diagram shows a paper cup in the shape of a cone.
The diameter of the top of the cup is 7 cm.
The volume of the cup is 110 cm 3 .
(a) Show that the height of the cup, h cm, is 8.57 correct to 2 decimal places.
[3]
(b) Calculate the slant height, l cm, of the cup.
l = ................................................. [2]
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(c)
A 7
NOT TO
SCALE
x°
O O
The cup is cut along the line OA.
It is opened out into a sector of a circle with centre O and sector angle x°.
Calculate the value of x.
x = ................................................. [4]
(d) A second paper cup is mathematically similar to the cup with volume 110 cm 3 .
The volume of the second cup is 165 cm 3 .
Calculate the diameter of the top of the second cup.
............................................ cm [2]
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7
(b)
Q
NOT TO
SCALE
O 96°
OPS and OQR are sectors of circles each with centre O.
OPQ and OSR are straight lines.
OP = 7.4 cm, PQ = 1.2 cm and QOR t = 96° .
Calculate the shaded area.
.......................................... cm2 [3]
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10 (a) A cuboid measures 6.2 cm by 4.8 cm by 2.5 cm.
Each measurement is given correct to the nearest millimetre.
Calculate the upper bound of the surface area of the cuboid.
.......................................... cm2 [3]
1
(b) [Volume of a pyramid = # base area # height]
3
19
17
D C
X
A B
The diagram shows a square-based pyramid ABCDE.
Vertex E is vertically above X, the centre of the square base.
The height of the pyramid, EX, is 17 cm.
EC = 19 cm.
(i) Show that the length of the base is 12 cm.
[4]
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(ii) Calculate the volume of the pyramid.
.......................................... cm3 [2]
(iii) Calculate angle CBE.
Angle CBE = ................................................ [3]
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1 2
9 [Volume of a cone = rr h]
3
[Curved surface area of a cone = rrl ]
A cone has radius 6 cm and slant height l cm.
The total surface area of the cone is 84r cm2.
(a) Show that l = 8.
[2]
(b) Calculate the volume of the cone.
.......................................... cm3 [3]
(c) A similar cone has a total surface area of 47.25r cm2.
Find the radius of this cone.
............................................ cm [2]
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(b)
P
NOT TO
8 SCALE
O 130° R
P and Q are points on the circumference of a different circle, centre O.
PR and QR are tangents to the circle at P and Q respectively.
OP = 8 cm and POQ t = 130° .
(i) Find PR.
PR = ........................................... cm [2]
(ii) Calculate the percentage of quadrilateral OPRQ that is shaded.
............................................. % [4]
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4
4 (a) [Volume of a sphere = rr 3 ]
3
[Surface area of a sphere = 4rr 2 ]
16
The diagram shows a solid formed by joining a cylinder to a hemisphere.
The diameter of the cylinder is 9 cm and its height is 16 cm.
(i) The volume of the hemisphere is equal to the volume of the cylinder.
Show that the radius of the hemisphere is 7.86 cm, correct to 2 decimal places.
[4]
(ii) Calculate the total surface area of the solid.
.......................................... cm 2 [3]
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10
5
H
D
G
2.25 E
C
1.85 F
A
2.10
1.55
B
The diagram shows a garden shed positioned on horizontal ground.
It is in the shape of a prism with trapezium ABCD as its cross-section.
The base of the shed, ABFE, is a rectangle.
AB = 1.55 m, AD = 2.25 m, BC = 1.85 m and BF = 2.10 m.
(a) Calculate the volume of the shed.
............................................ m 3 [3]
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(b) The roof of the shed, CGHD, is painted.
1 litre of paint covers 2 square metres.
Calculate the amount of paint used.
........................................ litres [4]
(c) Calculate the angle of elevation of D from F.
................................................. [4]
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(b)
Solid A
6
40°
The cross-section of solid A is the sector of a circle of radius 6 cm and angle 40°.
The height of solid A is 5 cm.
(i) Calculate the total surface area of solid A.
.......................................... cm 2 [4]
(ii) Solid B is mathematically similar to solid A.
The ratio volume of solid A : volume of solid B = 27 : 1.
Calculate the surface area of solid B.
.......................................... cm 2 [2]
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14
8 A birthday cake is in the shape of a cylinder.
There are two layers of cake and one layer of icing.
10 cm
3 cm
12 mm
3 cm
Each layer of cake has radius 10 cm and height 3 cm.
The icing, between the two layers of cake, has radius 10 cm and height 12 mm.
(a) Calculate the volume of icing in the birthday cake.
Give your answer in cm3.
........................................... cm3 [2]
(b) The top and curved surface of the birthday cake are now covered with chocolate.
Calculate the area of the birthday cake that is covered with chocolate.
.......................................... cm2 [3]
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(c) Anil has a slice of this chocolate-covered birthday cake.
10.3
x°
7.5
His slice is a prism of height 7.5 cm.
The top of the cake is a sector, radius 10.3 cm and angle x°.
The volume of his slice is 200 cm3.
Calculate the value of x.
x = ................................................ [3]
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8 (a)
L
NOT TO
SCALE
80
110°
45
A display notice is made by removing a sector of a circle from a larger sector.
Both sectors have an angle of 110°.
The radii of the sectors are 80 cm and 45 cm.
(i) Calculate arc length L.
L = ............................................ cm [2]
(ii) Calculate the area of this display notice.
.......................................... cm2 [3]
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(b)
NOT TO
SCALE
32 110°
This diagram shows a display notice mathematically similar to the one in part (a).
The radius of the larger sector is 32 cm.
Calculate the area of this display notice.
.......................................... cm2 [2]
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14
1
8 [Volume of cone = rr 2 h ]
3
[Curved surface area of a cone = rrl ]
16
15
12
c 45
The diagram shows a bowl with a circular base.
The curved surface of the bowl is formed by removing a cone with radius 12 cm and height 45 cm from a
larger cone as shown in the diagram.
The radius of the top of the bowl is 16 cm and its height is 15 cm.
(a) Calculate the volume of the bowl.
............................................. cm3 [3]
(b) The slant height of the cone that has been removed is c cm.
Show that c = 46.6, correct to 3 significant figures.
[2]
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(c) The bowl is completely filled with water.
Calculate the total surface area of the bowl that is in contact with the water.
............................................. cm2 [4]
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10 (a)
A D
NOT TO
SCALE
O
B C
AC and BD are diameters of the circle, centre O.
Show that triangle ABC is congruent to triangle BAD.
Give a reason for each statement you make.
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................
............................................................................................................................................................ [3]
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(b)
A D
NOT TO
SCALE
T
O
B C
Two tangents, TC and TD, are drawn to the circle in part (a).
t = 28°.
The diameter of the circle is 8 cm and ABD
(i) t .
Find COD
t = ................................................... [2]
COD
(ii) Calculate the area shown shaded in the diagram.
............................................. cm2 [5]
© UCLES 2019 4024/22/O/N/19
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1
4 [Volume of cone = rr 2 h ]
3
[Curved surface area of a cone = rrl ]
15
95
The diagram shows a gate post.
It is made in the shape of a cylinder with a cone on top.
The cylinder and the cone each have diameter 8 cm.
The height of the cylinder is 95 cm and the height of the cone is 15 cm.
(a) Calculate the volume of the gate post.
............................................. cm3 [3]
(b) Show that the total curved surface area of the gate post is 2580 cm2, correct to 3 significant figures.
[5]
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(c) A geometrically similar gate post has a total height of 150 cm.
Calculate the total curved surface area of this gate post.
............................................. cm2 [2]
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B
14
27°
A 15 C
The diagram shows a triangular prism.
AC = 15 cm, BC = 14 cm and angle ACB = 27°.
(a) Calculate AB.
AB = ............................................. cm [3]
(b) The length of the prism is p cm and the volume of the prism is 1000 cm3.
Calculate p.
p = ................................................... [3]
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(c) The prism is to be packed in a carton.
The carton is a cuboid of size 15 cm by p cm by h cm.
Calculate the smallest possible value of h.
h = ................................................... [2]
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16
4
9 (a) [Volume of a sphere = rr 3 ]
3
[Surface area of a sphere = 4rr 2 ]
24
The diagram shows lamp A.
It is made in the shape of a cylinder with a hemisphere on top.
The radius of the hemisphere and the radius of the cylinder are both 3 cm.
The total height of the lamp is 24 cm.
(i) Show that the volume of lamp A is 650 cm3, correct to 3 significant figures.
[4]
(ii) Calculate the curved surface area of lamp A.
Answer ................................... cm2 [3]
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(iii) Lamp B is mathematically similar to lamp A.
The volume of lamp B is 450 cm3.
Calculate the total height of lamp B.
Answer ..................................... cm [2]
(b) The mass of lamp C is 340 g, correct to the nearest 10 g.
8 of these lamps are placed in a packing case.
The total mass of the packing case and the 8 lamps is 4.2 kg, correct to the nearest 0.1 kg.
Calculate the upper bound of the mass of the packing case when empty.
Give your answer in kilograms.
Answer ..................................... kg [3]
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8
5 (a)
O
11
134°
A B
OAB is a sector of a circle, centre O, radius 11 cm.
t = 134° .
AOB
(i) Calculate the length of the arc AB.
Answer ..................................... cm [2]
(ii) Calculate the shortest distance from O to the line AB.
Answer ..................................... cm [2]
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1 2
(b) [Volume of a cone = rr h]
3
[Curved surface area of a cone = rrl]
9.5
A cone has height 9.5 cm and volume 115 cm3.
(i) Show that the radius of the base of the cone is 3.4 cm, correct to 1 decimal place.
[2]
(ii) Calculate the curved surface area of the cone.
Answer ....................................cm2 [3]
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16
70°
The diagram shows a sector of a circle of radius 8 cm and angle 70°.
(a) Calculate the shaded area.
Answer ................................... cm2 [4]
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(b)
16
A piece of chocolate is in the shape of a prism with the shaded area from part (a) being its
cross section.
The rectangular base of the chocolate is 16 cm by x cm.
The piece of chocolate is to be placed in a box which is a cuboid of size 16 cm by x cm by 1.5 cm.
(i) Show that the chocolate will fit inside the box.
[3]
(ii) These boxes are to be packed in cartons in the shape of a cuboid.
The size of each carton is 48 cm by 4x cm by 24 cm.
Find the maximum number of boxes that can be packed inside one carton.
Answer .......................................... [2]
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