The City School
CAMPUS
Mensuration
UNIT 9LIAQUAT
C MATHEMATICS
2.5Mensuration
Worksheet extracted from Past Paper Questions
Perimeter and Area of Figures
1.
Figure
Diagram
Square
l
Formulae
Area = l2
Perimeter = 4l
l
b
Rectangle
Area = l × b
Perimeter = 2(l + b)
l
Triangle
1
× base × height
2
1
=
×b×h
2
Area =
h
b
Parallelogram
Area = base × height
=b×h
h
b
a
Trapezium
h
Area =
1
(a + b) h
2
b
h
Rhombus
Area = b × h
b
Mensuration
129
1
Figure
Diagram
Formulae
Area = πr2
Circumference = 2πr
r
Circle
r
Annulus
Area = π(R2 – r2)
R
θ°
× 2πr
360°
(where θ is in degrees)
= rθ (where θ is in radians)
s
Sector
Arc length s =
r
θ
θ°
× πr2 (where θ is in degrees)
360°
1
= r2θ (where θ is in radians)
2
Area =
O
s
Segment
θ
Area =
r
1 2
r (θ – sin θ)
2
(where θ is in radians)
Volume andO Surface Area of Solids
5.
Figure
Figure
Diagram
Sphere
Formulae
4 3
πr
3
Surface area = 4πr 2
r
Volume =
Cuboid
h
r
Hemisphere
Formulae
Diagram
Volume =
2 3
πr
3
Volume = l × b × h
Total surface area = 2(lb + l
2
2
Surface
b area = 2πr + πr
l
= 3πr2
r
Volume = πr2h
Cylinder
Curved surface area = 2πrh
h
(a) 2.5
A sphere has a radius of 10 cm. Calculate the volume of the sphere.Total surface area = 2πrh +
UNIT
Example 3
130
(b) A cuboid has the same volume as the sphere in part (a). The length and
breadth of the cuboid are both 5 cm. Calculate the height of the cuboid.
Leave your answers in terms of π.
Solution
4π(10)3
(a) Volume =
3
Prism = 4000 π cm3
3
Volume
= Area of cross
section × len
2
Total surface area
Volume and Surface Area of Solids
5.
Figure
Diagram
Formulae
Cuboid
h
Volume = l × b × h
Total surface area = 2(lb + lh + bh)
b
l
r
h
Cylinder
Volume = πr2h
Curved surface area = 2πrh
Total surface area = 2πrh + 2πr2
Volume
= Area of cross section × length
Total surface area
= Perimeter of the base × height
+ 2(base area)
Prism
h
Pyramid
Volume =
1
× base area × h
3
1 2
πr h
3
Curved surface area = πrl
Volume =
Cone
l
h
(where l is the slant height)
r
134
Total surface area = πrl + πr2
UNIT 2.5
3
9
9
(a) The diagram shows a roll of material.
The material is wound onto a metal
cylinder whose cross-section is a circle
of radius 10 cm.
The shaded area shows the cross-section
of the material on the roll.
The outer layer of material forms the
curved surface of a cylinder of
radius 30 cm.
30
10
200
(i) Calculate, in square centimetres, the area of the cross-section of the material on the roll (shaded
on the diagram).
[2]
(ii) The material is 200 cm wide on the roll.
Calculate, in cubic metres, the volume of the material.
[2]
(iii) When unwound, the length of the material is 150 m.
Calculate the thickness of the material, giving your answer in millimetres.
[2]
(b) The diagram shows a conical tent.
The diameter of the base is 3.5 m and the slant height is 3 m.
It is made from a flat piece of canvas that forms a sector of a circle of
radius 3 m. The angle at the centre is !°.
3
3.5
3
θ°
(i) Show that ! = 210.
[3]
(ii) As shown, the required shape is cut from a rectangular piece of canvas of width w metres.
w
Given that w is a whole number, find its least possible value.
Show all your working.
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6
For
Examiner’s
Use
11
A rectangular box has dimensions 30 cm by 10 cm by 5 cm.
A container holds exactly 100 of these boxes.
(a) Calculate the total volume, in cubic metres, of the 100 boxes.
(b) Each box has a mass of 1.5 kg to the nearest 0.1 kg.
The empty container has a mass of 6 kg to the nearest 0.5 kg.
Calculate the greatest possible total mass of the container and 100 boxes.
Answer (a) ................................... m3 [1]
(b) .................................... kg [2]
12 Given that f (x) = 4x + 3 , find
2x
(a) f(3) ,
(b) f –1 (x) .
Answer (a) f(3) = ..................................[1]
(b) f –1 (x) = ..............................[2]
© UCLES 2008
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Examiner’s
Use
7
Section B [48 marks]
Answer four questions in this section.
Each question in this section carries 12 marks.
7
A, B, C, D and E are five different shaped blocks of ice stored in a refrigerated room.
(a) At 11 p.m. on Monday the cooling system failed, and the blocks started to melt.
At the end of each 24 hour period, the volume of each block was 12% less than its volume at the start
of that period.
(i) Block A had a volume of 7500 cm3 at 11 p.m. on Monday.
Calculate its volume at 11 p.m. on Wednesday.
[2]
(ii) Block B had a volume of 6490 cm3 at 11 p.m. on Tuesday.
Calculate its volume at 11 p.m. on the previous day.
[2]
(iii) Showing your working clearly, find on which day the volume of Block C was half its volume at
11 p.m. on Monday.
[2]
(b) [The volume of a sphere is 43πr 3.]
[The surface area of a sphere is 4πr 2.]
At 11 p.m. on Monday Block D was a hemisphere with radius 18 cm.
Calculate
(i) its volume,
[2]
(ii) its total surface area.
[2]
(c) As Block E melted, its shape was always geometrically similar to its original shape.
It had a volume of 5000 cm3 when its height was 12 cm.
Calculate its height when its volume was 1080 cm3.
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For
Examiner’s
Use
For
Examiner’s
Use
4
7
The diagram shows a solid cuboid with base 10 cm by 6 cm.
The height of the cuboid is x centimetres.
x
6
10
(a) Find an expression, in terms of x, for the total surface area of the cuboid.
(b) The total surface area of the cuboid is 376 cm2.
Form an equation in x and solve it to find the height of the cuboid.
Answer (a) ................................. cm2 [1]
(b) ................................. cm
8
[2]
Evaluate
(a) 90,
(b) 9–2,
3
2
(c) 9 .
Answer (a) ............................................[1]
(b) ............................................[1]
(c) ............................................[1]
© UCLES 2008
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6
Section B [48 marks]
Answer four questions in this section.
Each question in this section carries 12 marks.
7
(a)
A fuel tank is a cylinder of diameter 1.8 m.
(i) The tank holds 25 000 litres when full.
Given that 1m3 = 1000 litres, calculate the length of the cylinder.
Give your answer in metres.
(ii)
[4]
A
O
C
0.9
E
D
B
fuel
The diagram shows the cross-section of the cylinder, centre O, containing some fuel.
CD is horizontal and is the level of the fuel in the cylinder.
AB is a vertical diameter and intersects CD at E.
Given that E is the midpoint of OB,
(a) show that EÔD = 60°,
[1]
(b) calculate the area of the segment BCD,
[3]
(c) calculate the number of litres of fuel in the cylinder.
[2]
4 3
πr ]
3
A different fuel tank consists of a cylinder of diameter 1.5 m
and a hemisphere of diameter 1.5 m at one end.
(b) [Volume of a sphere =
1.5
1.5
The volume of the cylinder is 10 times the volume of the hemisphere.
Calculate the length of the cylinder.
© UCLES 2009
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7
Section B [48 marks]
Answer four questions in this section.
Each question in this section carries 12 marks.
7
(a) When a solid rectangular wooden block of oak floats, 60% of its height is under water.
(i) What fraction of its height is above water?
[1]
(ii) A block of oak has length 60 cm,
breadth 50 cm and height
h centimetres.
15
h
It floats with 15 cm of its height
under water.
60
50
(a) Find the value of h.
[1]
(b) In the diagram, the shaded region represents part of the surface area of the block that is in
contact with the water.
Calculate the total surface area of the block that is in contact with the water.
(b)
[2]
35
A
A
B
O
220°
B
9
A solid cylinder, made from a different type of wood, floats in water.
The shaded region represents part of the surface of the cylinder that is in contact with the water.
The right hand diagram shows the circular cross-section of one end.
The centre of the circle is O and the water level reaches the points A and B on the circumference.
Reflex angle AOB = 220° .
The cylinder has radius 9 cm and length 35 cm.
Calculate
(i) the area of the curved surface of the cylinder that is in contact with the water,
[2]
(ii) the surface area of one end of the cylinder that is in contact with the water,
[4]
(iii) the distance between the water level AB and the top of the cylinder.
[2]
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19
(b) For the remaining piece of glass ABCDGF, find
For
Examiner’s
Use
(i) its perimeter,
Answer (b)(i) ......................... cm [2]
(ii) its area.
Answer (b)(ii) ....................... cm2 [2]
(c) State the value of cos DĜF.
Answer (c) ...................................... [1]
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18
22
A
For
Examiner’s
Use
30
F
13
30
D
G
5
E
15
B
C
ABCDEF represents an L-shaped piece of glass with AB = AF = 30 cm and CD = 15 cm.
The glass is cut to fit the window in a door and the shaded triangle DEG is removed.
DG = 13 cm and EG = 5 cm.
(a) Show that DE = 12 cm.
Answer (a) .................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
.............................................................................................................................................. [1]
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9
9
B
C
B
5
C
A 5
5
D
D
Diagram II
Diagram I
Diagram I shows a cube with a triangular pyramid removed from one vertex.
This triangular pyramid ABCD is shown in Diagram II.
AB = AC = AD = 5 cm.
(a) State the height of this pyramid when the base is triangle ABD.
[1]
(b) [The volume of a pyramid = 1 × area of base × height]
3
Calculate
(i)
the volume of the pyramid,
[2]
(ii)
the area of triangle BCD,
[3]
(iii)
the height of the pyramid when the base is triangle BCD.
[3]
(c) An identical triangular pyramid is removed from each of the other 7 vertices of the cube to form
the new solid shown in Diagram III.
Diagram III
The original cube had 6 faces, 8 vertices and 12 edges.
For the new solid, write down the number of
(i)
faces,
[1]
(ii)
vertices,
[1]
(iii)
edges.
[1]
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12
1
12 [Volume of a cone = 3 π r 2h]
[Curved surface area of a cone = π rl]
B
Diagram I shows a solid cone with C
as the centre of its base.
B is the vertex of the cone and A is a point
on the circumference of its base.
AC = 9 cm and BC = 12 cm.
Diagram I
12
A
C
9
(a) Calculate
(i) AB,
[2]
(ii) the total surface area of the cone,
[2]
(iii) the volume of the cone.
[2]
(b) The cone in Diagram I is cut, parallel
to the base, to obtain a small cone shown
in Diagram II and a frustum shown in
Diagram III.
Y is the centre of the base of the small cone.
X is the point on the circumference of this
base and on the line AB such that
XY = 3 cm.
B
Diagram II
Diagram III
A
X
3
X
3 Y
Y
C
Calculate
(i) BY,
[1]
(ii) AX,
[1]
(iii) the circumference of the base of the small cone,
[2]
(iv) the volume of the frustum.
[2]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been
made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at
the earliest possible opportunity.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
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7
7
(a)
17
P
Q
8
S
R
29
PQRS is a trapezium.
PQ = 17 cm, QR = 8 cm, SR = 29 cm and SR̂Q = 90°.
Calculate
(i) the area of PQRS,
[1]
(ii) PŜR.
[2]
(b)
L
18
15
10
K
N
M
In the diagram, triangle KLM is similar to triangle LNM.
KL = 15 cm, LM = 18 cm and LN = 10 cm.
(i) Find KM.
[2]
(ii) Find KN.
[2]
(iii) P is the point on LM such that PN is parallel to LK.
Find
the area of triangle NPM .
the area of trapezium KLPN
Give your answer as a fraction in its simplest form.
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[2]
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13
A
18 OAB is the sector of a circle of radius r cm.
AÔB = 60°.
For
Examiner’s
Use
Find, in its simplest form, an expression in terms of r and π for
60°
O
(a) the area of the sector,
r
B
Answer (a) .............................. cm2 [1]
(b) the perimeter of the sector.
Answer (b) ............................... cm [2]
19 A =
!–13 12" and B = !–10 23" .
Find
(a) A – B,
Answer (a)
!
"
[1]
Answer (b)
!
"
[2]
(b) B–1.
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21
(ii) Find the circumference of the base of the cone.
Do not
write in this
margin
Answer .................................. cm [2]
(iii) The curved surface area of the cone
can be made into the shape of a
sector of a circle with angle θº.
18
θ°
Show that θ is 200, correct to
the nearest integer.
18
[2]
(iv) Hence, or otherwise, find the total surface area of the solid.
Answer .................................cm2 [3]
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20
11
[Volume of a cone =
1 2
π r h]
3
Do not
write in this
margin
r
H
r
The solid above consists of a cone with base radius r centimetres on top of a cylinder
of radius r centimetres.
The height of the cylinder is twice the height of the cone.
The total height of the solid is H centimetres.
(a) Find an expression, in terms of π, r and H, for the volume of the solid.
Give your answer in its simplest form.
Answer ....................................... [3]
(b) It is given that r = 10 and the height of the cone is 15 cm.
(i) Show that the slant height of the cone is 18.0 cm, correct to one decimal place.
[2]
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4
2
(a) The formula for the area of a trapezium is
A = 12 h ( c + d ).
(i) Find an expression for c in terms of A, h and d.
Answer ........................................ [2]
(ii)
c
4
8
The diagram shows a trapezium with dimensions given in centimetres.
The perpendicular distance between the parallel lines is 4 cm.
The area of the trapezium is 22 cm2.
Find c.
Answer ........................................ [1]
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4
5
c=
!32"
d=
!–86 "
(a) Calculate 2c – d .
Answer
! "
Answer
........................................ [1]
[1]
(b) Calculate # d # .
6
A
6
B
9
C
ABC is a right-angled triangle with AB = 6 cm and BC = 9 cm.
A semicircle of diameter 6 cm is joined to the triangle along AB.
Find an expression, in the form a + bπ, for the total area of the shape.
Answer
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................................. cm2 [2]
18
12 [The volume of a sphere = 43 π r3]
Do not
write in this
margin
x
x
x
A solid consists of a sphere on top of a square-based cuboid.
The diameter of the sphere is x cm.
The base of the cuboid has sides of length x cm.
The sum of the height of the cuboid and one of the sides of the base is 8 cm.
(a) By considering the height of the cuboid, explain why it is not possible for this sphere to
have a radius of 5 cm.
Answer ......................................................................................................................................
............................................................................................................................................. [1]
(b) By taking the value of π as 3, show that the approximate volume, y cm3, of the solid is given by
y = 8x 2 –
x3
.
2
[2]
(c) The table below shows some values of x and the corresponding values of y for
x
1
2
y
7.5
28
y = 8x 2 –
x3
.
2
3
4
5
6
7
96
137.5
180
220.5
(i) Complete the table.
[1]
(ii) On the grid opposite, plot the graph of y = 8x 2 –
x3
for 1 ! x ! 7.
2
[3]
(iii) Use your graph to find the height of the cuboid when the volume of the solid is 120 cm3.
Answer ................................. cm [2]
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15
(b) Shape I is formed by joining this segment
to a trapezium, ABCD, along AB.
AB is parallel to DC, DC = 25 cm and the
perpendicular height of the trapezium is h cm.
The area of the trapezium is 248 cm2.
Do not
write in this
margin
60° 15
Calculate h.
B
A
h
D
25
Shape I
C
Answer ........................................ [2]
(c) Shape II is geometrically similar to Shape I.
The longest side of the trapezium in Shape II is 5 cm.
r
5
Shape II
(i) Find the radius, r, of the segment in Shape II.
Answer ................................. cm [1]
(ii) Find the total area of Shape II.
Answer ................................ cm2 [3]
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4
5
y is inversely proportional to the square of x.
For
Examiner’s
Use
Given that y = 2 when x = 6, find the value of y when x = 2.
Answer y = ............................... [2]
6
A circle of diameter 6 cm is cut from a square of side 8 cm.
Find an expression, in the form a – bπ , for the shaded area.
6
Answer
© UCLES 2012
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8
..............................cm2 [2]
3
3
The diagram shows an L-shaped piece of card.
For
Examiner’s
Use
3
The measurements are in centimetres and
all the angles are right-angles.
4
8
(a) Calculate the perimeter of this card.
9
Answer
............................... cm [1]
(b) Square pieces, each of side 2 cm, are cut from this card.
Find the greatest number of squares that can be obtained.
4
Answer
..................................... [1]
Answer
..................................... [1]
f(x) = 5 + 3x
(a) Evaluate f !–
1
.
2
"
(b) Find f –1(x).
Answer f –1(x) = ........................ [1]
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15
4
[The Surface area of a sphere is 4πr2] [The Volume of a sphere is πr3]
3
Do not
write in this
margin
(b) A circular top that can hold 4 hemispherical bowls can be placed on the container.
7
7
21
Container and Top
8
Top
Cross-section
The top is a circle of diameter 21 cm with four circular holes of diameter 7 cm.
A hemispherical bowl of diameter 7 cm fits into each hole.
The cross-section shows two of these bowls.
(i) Calculate the inside curved surface area of one of these hemispherical bowls.
Answer
............................. cm2 [1]
(ii) Calculate the total surface area of the top of the container, including the inside curved
surface area of each bowl.
Answer
............................. cm2 [3]
(iii) With the top and the 4 bowls in place, calculate the volume of water required to fill the
container.
Answer
© UCLES 2012
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..............................cm3 [3]
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14
Section B [48 marks]
Do not
write in this
margin
Answer four questions in this section.
Each question in this section carries 12 marks.
7
A cylindrical, open container has a
diameter of 21 cm and height of 8 cm.
21
(a) (i) Calculate the total external surface area
of this container.
8
Answer
............................. cm2 [3]
(ii) A manufacturer receives an order for 30 000 containers.
He needs an extra 150 cm2 of material for each container to cover wastage.
Calculate the area of material needed to make these containers.
Give your answer in square metres.
Answer
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.............................. m2 [2]
11
(b) Calculate the area of the shaded shape ABCD.
Do not
write in this
margin
Answer
............................. cm2 [3]
(c) The shape ABCD is used to make a lampshade by joining AB and DC.
A,D
r
B,C
Calculate the radius, r cm, of the circular top of the lampshade.
Answer
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............................. cm [2]
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10
5
B
A
25
20
Do not
write in this
margin
O
150°
D
C
A
AD and BC are arcs of circles with centre O.
A is a point on OB, and D is a point on OC.
OA = 20 cm and AB = 25 cm.
AÔD = 150°.
(a) Calculate the perimeter of the shaded shape ABCD.
Answer
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.............................. cm [3]
15
(b) The total area of the metal (unshaded) sections of the cover is
55
π cm2.
3
For
Examiner’s
Use
(i) Calculate the total area of the shaded sections, giving your answer in terms of π.
Answer
.....................................cm2 [1]
(ii) Calculate the fraction of the total area of the cover that is metal (unshaded).
Give your answer in its simplest form.
Answer
............................................ [1]
Answer
............................................ [1]
Answer
............................................ [1]
Answer
............................................ [1]
20 (a) Evaluate
(i) 50 + 52,
(ii)
(iii)
1
36 2 ,
^2 3h .
2
6
1 k
(b) c m = 9
3
Find the value of k.
Answer k = ...................................... [1]
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16
9
(a) Shape Iisacylinderwithradius4cmandheighthcm.
ThevolumeofShape Iis1500cm3.
For
Examiner’s
Use
4
(i) Findh.
h
Shape I
Answer ............................................... [2]
(ii) Shape Iismadebypouringliquidintoamouldatarateof0.9litresperminute.
Findthenumberofsecondsittakestopourthisliquidintothemould.
Answer ................................. seconds[1]
(b) Shape IIisaprismoflength8cmwithatriangularcross-section,
shownshaded.
Twosidesoftheshadedtriangleareatrightanglestoeachotherand
havelengths5xcmand12xcm.
12x
GiventhatShape IIalsohasavolumeof1500cm3,findx.
5x
8
Shape II
Answer ............................................... [2]
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12
16 Maryam makes two geometrically similar cakes.
The heights of the cakes are 6 cm and 9 cm.
For
Examiner’s
Use
(a) Maryam decorates each cake with a ribbon around the outside.
The length of the ribbon for the larger cake is 66 cm.
Find the length of the ribbon for the smaller cake.
Answer
...................................... cm [1]
(b) Maryam uses 1600 m3 of cake mixture to make the smaller cake.
Find the volume of cake mixture she uses to make the larger cake.
17
p = 2.4 × 102
Answer
.....................................cm3 [2]
Answer
............................................ [1]
Answer
............................................ [2]
q = 6 × 103
Giving your answers in standard form, find
(a) p + q,
(b) 2p ÷ q.
© UCLES 2013
4024/11/M/J/13
2
ELECTRONIC CALCULATORS MUST NOT BE USED IN THIS PAPER.
1
In this shape all the lengths are in centimetres.
5
20
10
12
25
5
Work out
(a) the perimeter,
Answer
...................................... cm [1]
Answer
.....................................cm2 [1]
Answer
............................................ [1]
Answer
............................................ [1]
(b) the area.
2
Evaluate
(a) 0.3 × 0.2,
(b) 3.5 ÷ 0.07 .
© UCLES 2013
4024/11/M/J/13
For
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Use