Worksheet – Practice Questions

1. The base of a prism is a regular hexagon. The centre of the hexagon is O and the length of OA
is 15 cm.

diagram not to scale

(a) Write down the size of angle AOB.

(1)

(b) Find the area of the triangle AOB.

(3)

The height of the prism is 20 cm.

(c) Find the volume of the prism.

(2)
(Total 6 marks)

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2. The diagram shows a circle of radius R and centre O. A triangle AOB is drawn inside the circle.
The vertices of the triangle are at the centre, O, and at two points A and B on the circumference.
Angle AÔB is 110 degrees.

A B

R 110° R

(a) Given that the area of the circle is 36 cm2, calculate the length of the radius R.

(b) Calculate the length AB.

(c) Write down the side length L of a square which has the same area as the given circle.
(Total 6 marks)

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3. The diagram below shows a field ABCD with a fence BD crossing it. AB = 15 m, AD = 20 m
and angle BÂD = 110°. BC = 22 m and angle BD̂C = 30°.

A
diagram not to scale
15 110° 20

B D
30°
22

(a) Calculate the length of BD.

(3)

(b) Calculate the size of angle BĈD.

(3)

One student gave the answer to (a) “correct to 1 significant figure” and used this answer to
calculate the size of angle BĈD.

(c) Write down the length of BD correct to 1 significant figure.

(1)

(d) Find the size of angle BĈD that the student calculated, giving your answer correct to
1 decimal place.
(2)

(e) Hence find the percentage error in his answer for angle BĈD.
(3)
(Total 12 marks)

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4. A dog food manufacturer has to cut production costs. She wishes to use as little aluminium as
possible in the construction of cylindrical cans. In the following diagram, h represents the height
of the can in cm, and x represents the radius of the base of the can in cm.

diagram not to scale

The volume of the dog food cans is 600 cm3.

600
(a) Show that h = .
πx 2
(2)

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(b) (i) Find an expression for the curved surface area of the can, in terms of x.
Simplify your answer.

(ii) Hence write down an expression for A, the total surface area of the can, in terms of
x.
(4)

(c) Differentiate A in terms of x.

(3)

(d) Find the value of x that makes A a minimum.

(3)

(e) Calculate the minimum total surface area of the dog food can.
(2)
(Total 14 marks)

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5. A child’s toy is made by combining a hemisphere of radius 3 cm and a right circular cone of
slant height l as shown on the diagram below.

diagram not to scale

(a) Show that the volume of the hemisphere is 18 cm3.

(2)

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 The volume of the cone is two-thirds that of the hemisphere.

(b) Show that the vertical height of the cone is 4 cm.

(4)

(c) Calculate the slant height of the cone.

(2)

(d) Calculate the angle between the slanting side of the cone and the flat surface of the
hemisphere.
(3)

(e) The toy is made of wood of density 0.6 g per cm3. Calculate the weight of the toy.
(3)

(f) Calculate the total surface area of the toy.

(5)
(Total 19 marks)

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6. The diagram below shows a child’s toy which is made up of a circular hoop, centre O, radius 7
cm. The hoop is suspended in a horizontal plane by three equal strings XA, XB, and XC. Each
string is of length 25 cm. The points A, B and C are equally spaced round the circumference of
the hoop and X is vertically above the point O.

diagram not to scale

(a) Calculate the length of XO.

(2)

(b) Find the angle, in degrees, between any string and the horizontal plane.
(2)

(c) Write down the size of angle AÔB.

(1)

(d) Calculate the length of AB.

(3)

(e) Find the angle between strings XA and XB.

(3)
(Total 11 marks)

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7. ABCDV is a solid glass pyramid. The base of the pyramid is a square of side 3.2 cm. The
vertical height is 2.8 cm. The vertex V is directly above the centre O of the base.

D C

A B

(a) Calculate the volume of the pyramid.

(2)

(b) The glass weighs 9.3 grams per cm3. Calculate the weight of the pyramid.
(2)

(c) Show that the length of the sloping edge VC of the pyramid is 3.6 cm.
(4)

(d) Calculate the angle at the vertex, BV̂C .

(3)

(e) Calculate the total surface area of the pyramid.

(4)
(Total 15 marks)

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8. Jenny has a circular cylinder with a lid. The cylinder has height 39 cm and diameter 65 mm.

(a) Calculate the volume of the cylinder in cm3. Give your answer correct to two decimal
places.
(3)

The cylinder is used for storing tennis balls.

Each ball has a radius of 3.25 cm.

(b) Calculate how many balls Jenny can fit in the cylinder if it is filled to the top.
(1)

(c) (i) Jenny fills the cylinder with the number of balls found in part (b) and puts the lid
on. Calculate the volume of air inside the cylinder in the spaces between the tennis
balls.

(ii) Convert your answer to (c) (i) into cubic metres.

(4)
(Total 8 marks)

9. The diagram shows an office tower of total height 126 metres. It consists of a square-based
pyramid VABCD on top of a cuboid ABCDPQRS.

V is directly above the centre of the base of the office tower.

The length of the sloping edge VC is 22.5 metres and the angle that VC makes with the base
ABCD (angle VCA) is 53.1°.

diagram not to scale

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(a) (i) Write down the length of VA in metres.

(ii) Sketch the triangle VCA showing clearly the length of VC and the size of angle
VCA.
(2)

(b) Show that the height of the pyramid is 18.0 metres correct to 3 significant figures.
(2)

(c) Calculate the length of AC in metres.

(3)

(d) Show that the length of BC is 19.1 metres correct to 3 significant figures.
(2)

(e) Calculate the volume of the tower.

(4)

To calculate the cost of air conditioning, engineers must estimate the weight of air in the tower.
They estimate that 90 % of the volume of the tower is occupied by air and they know that 1 m3
of air weighs 1.2 kg.

(f) Calculate the weight of air in the tower.

(3)
(Total 16 marks)

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10. (a) A gardener has to pave a rectangular area 15.4 metres long and 5.5 metres wide using
rectangular bricks. The bricks are 22 cm long and 11 cm wide.

(i) Calculate the total area to be paved. Give your answer in cm2.

(ii) Write down the area of each brick.

(iii) Find how many bricks are required to pave the total area.
(6)

(b) The gardener decides to have a triangular lawn ABC, instead of paving, in the middle of
the rectangular area, as shown in the diagram below.

diagram not to scale

The distance AB is 4 metres, AC is 6 metres and angle BAC is 40°.

(i) Find the length of BC.

(ii) Hence write down the perimeter of the triangular lawn.

(iii) Calculate the area of the lawn.

(iv) Find the percentage of the rectangular area which is to be lawn.

(9)

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(c) In another garden, twelve of the same rectangular bricks are to be used to make an edge
around a small garden bed as shown in the diagrams below. FH is the length of a brick
and C is the centre of the garden bed. M and N are the midpoints of the long edges of the
bricks on opposite sides of the garden bed.

diagram not to scale

(i) Find the angle FCH.

(ii) Calculate the distance MN from one side of the garden bed to the other, passing
through C.
(5)

The garden bed has an area of 5419 cm2. It is covered with soil to a depth of 2.5 cm.

(d) Find the volume of soil used.

(2)

It is estimated that 1 kilogram of soil occupies 514 cm3.

(e) Find the number of kilograms of soil required for this garden bed.
(2)
(Total 24 marks)

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