Worksheet 1 Topic: Mensuration

1 (a)
NOT TO
SCALE
16 cm 1.5 cm

The diagram shows a solid made from a cylinder and a cone.

The height of the cylinder is 16 cm and the height of the cone is 1.5 cm.
The radius of the cylinder and the base radius of the cone are each 0.35 cm.

(i) Calculate the total surface area of the solid.

[The curved surface area, A, of a cone with radius r and slant height l is A = rrl .]

......................................... cm 2 [5]

(ii) Calculate the volume of the solid.

1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3

......................................... cm 3 [3]

(iii)
NOT TO
1.4 cm SCALE

3.5 cm

10 of the solids are placed in a box in the shape of a cuboid of length 17.5 cm.
The diagram shows one end of the box.

Calculate the volume of the empty space in the box.

......................................... cm 3 [3]
ĬÍĉ¿Ġ³íËñÑāÍĢÑĐÆĤ×
© UCLES 2024 ĬĚĘã¾Ğ¾ćÖÿ÷Ě¼ĆĎĊàĂ
ĥĕĥÕµõåõÕąÕĕõåĥĕĥÕ
0580/43/M/J/24
 2

(b)

NOT TO
SCALE

The diagram shows two mathematically similar solids.

The surface area of the larger solid is 200 cm 2 and the surface area of the smaller solid is 98 cm 2 .
The volume of the larger solid is 450 cm 3 .

Calculate the volume of the smaller solid.

......................................... cm 3 [3]

© UCLES 2024 0580/43/M/J/24 [Turn over

 3

2 (a)

NOT TO
12 cm SCALE
1m

The diagram shows a tank in the shape of a half-cylinder of radius 12 cm and length 1 metre.
The tank is fixed horizontally and is completely filled with water.

(i) Calculate the volume of water in the tank.

Give your answer correct to the nearest 10 cm 3 .

.......................................... cm 3 [3]

(ii)
NOT TO
6 cm SCALE

Water is removed from the tank until the level of water is 6 cm below the top of the tank.
The diagram shows the cross-section of the tank.

Calculate the volume of water that is now in the tank.

.......................................... cm 3 [5]

© UCLES 2024 0580/42/M/J/24

 4

(b) A rectangular fish tank with length 42 cm and width 35 cm is full of water.
A stone lies at the bottom of the tank.
When the stone is removed from the tank, the depth of the water decreases by 0.2 cm.
The density of the stone is 2.2 g/cm 3 .

Calculate the mass of the stone in grams.

[Density = mass ' volume]

............................................... g [3]

(c)
H G

E
F

15 cm
NOT TO
SCALE

D
C
12 cm
A 8 cm B

The diagram shows a cuboid, ABCDEFGH.

Calculate the angle that AG makes with the base of the cuboid.

................................................. [4]
© UCLES 2024 0580/42/M/J/24 [Turn over
 5

3 (a)
NOT TO
SCALE
40 cm

30 cm
70 cm

The diagram shows a box in the shape of a cuboid.

The box is open at the top.

(i) Work out the surface area of the inside of the open box.

......................................... cm 2 [3]

(ii) Cylinders with height 20 cm and diameter 15 cm are placed in the box.

Work out the maximum number of these cylinders that can completely fit inside the box.

................................................. [3]
© UCLES 2024 0580/41/M/J/24
 6

(b) A solid bronze cone has a mass 750 g.

The density of the bronze is 8.9 g/cm 3 .

The ratio radius of cone : height of cone = 1 : 3.

(i) Show that the radius of the cone is 2.99 cm, correct to 3 significant figures.
[Density = mass ÷ volume]
1
[The volume, V , of a cone with radius r and height h is V = rr 2 h .]
3

[4]

(ii) Calculate the total surface area of the cone.

[The curved surface area, A, of a cone with radius r and slant height l is A = rrl .]

......................................... cm 2 [5]

© UCLES 2024 0580/41/M/J/24 [Turn over

 7

4 (a)

NOT TO
15 cm
SCALE

8 cm

A cone has base diameter 8 cm and perpendicular height 15 cm.

(i) Calculate the volume of the cone.

1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3

......................................... cm 3 [2]

(ii) A label completely covers the curved surface area of the cone.

Calculate the area of the label as a percentage of the total surface area of the cone.
[The curved surface area, A, of a cone with radius r and slant height l is A = rrl .]

............................................. % [5]

© UCLES 2023 0580/42/M/J/23

 8

(b)

NOT TO
SCALE
0.45 m

An empty cylindrical container has radius 0.45 m.

300 litres of water is poured into the container at a rate of 375 ml per second.

(i) Find the time taken, in minutes and seconds, for all the water to be poured into the container.

............................. min ............................. s [3]

(ii) Calculate the height of the water in the container.

............................................. m [3]

© UCLES 2023 0580/42/M/J/23 [Turn over

 9

5 (a)
P NOT TO
SCALE
8 cm

Q R
24 cm

(i) Calculate the area of triangle PQR.

......................................... cm 2 [2]

(ii) Calculate angle PRQ.

Angle PRQ = ................................................ [2]

(b)

NOT TO
SCALE

11 cm

6 cm

The diagram shows a half-cylinder of radius 6 cm and length 11 cm.

Calculate the volume of the half-cylinder.

......................................... cm 3 [2]

© UCLES 2023 0580/41/M/J/23

 10

(c)
D T C T C
4 cm
S S
O
15 cm NOT TO
X X SCALE

A 20 cm B A B

(i) ABCD is a rectangle with AB = 20 cm and BC = 15 cm.

S, X and T are points on a circle centre O, such that DSA and DTC are tangents to the circle.
The radius of the circle is 4 cm and TX is a diameter of the circle.
The shape DSXT is removed from the corner of the rectangle, leaving the shaded shape shown
in the second diagram.

Calculate the area of the shaded shape.

......................................... cm 2 [5]

(ii) Calculate the perimeter of the shaded shape.

............................................ cm [3]

© UCLES 2023 0580/41/M/J/23 [Turn over

 12

7 (a)

A 28 cm D AD
NOT TO
SCALE

20 cm

N BC
B C

A rectangular sheet of paper ABCD is made into an open cylinder with the edge AB meeting the
edge DC.
AD = 28 cm and AB = 20 cm.

(i) Show that the radius of the cylinder is 4.46 cm, correct to 3 significant figures.

[2]

(ii) Calculate the volume of the cylinder.

......................................... cm 3 [2]

(iii) N is a point on the base of the cylinder, such that BN is a diameter.

Calculate the angle between AN and the base of the cylinder.

................................................. [3]
© UCLES 2022 0580/42/M/J/22
 13

(b) The volume of a solid cone is 310 cm 3 .

The height of the cone is twice the radius of its base.

Calculate the slant height of the cone.

1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3

............................................ cm [5]

© UCLES 2022 0580/42/M/J/22 [Turn over

 14

8 (a) 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

NOT TO
SCALE

(i) (a) Show that angle OAM is 67.5°.

[2]

(b) Calculate the area of the octagon.

.......................................... cm 2 [4]

(ii) Find the area of the circle that passes through the vertices of the octagon.

.......................................... cm 2 [3]
© UCLES 2022 0580/41/M/J/22
 15

(b)
NOT TO
SCALE

4m
0.45 m

The diagram shows a horizontal container for water with a uniform cross-section.
The cross-section is a semicircle.
The radius of the semicircle is 0.45 m and the length of the container is 4 m.

(i) Calculate the volume of the container.

........................................... m 3 [2]

(ii)
NOT TO
SCALE

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]

© UCLES 2022 0580/41/M/J/22 [Turn over
 16

9 (a) A solid cuboid measures 20 cm by 12 cm by 5 cm.

(i) Calculate the volume of the cuboid.

.......................................... cm3 [1]

(ii) (a) Calculate the total surface area of the cuboid.

.......................................... cm2 [3]

(b) The surface of the cuboid is painted.

The cost of the paint used is $1.52 .

Find the cost to paint 1cm 2 of the cuboid.

Give your answer in cents.

........................................ cents [1]

9x
(b) A solid metal cylinder with radius x and height is melted.
2
All the metal is used to make a sphere with radius r.

Find r in terms of x.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3

r = ................................................. [3]

© UCLES 2021 0580/43/M/J/21

 17

(c)

NOT TO
SCALE

20 cm

5 cm 150 cm

The diagram shows a cylinder of length 150 cm on horizontal ground.

The cylinder has radius 20 cm.
The cylinder contains water to a depth of 5 cm, as shown in the diagram.

Calculate the volume of water in the cylinder.

Give your answer in litres.

........................................ litres [7]

© UCLES 2021 0580/43/M/J/21 [Turn over

 20

11 (a)

NOT TO
SCALE
6.3 cm

R cm

2.4 cm

The diagram shows a solid cone and a solid hemisphere.

The cone has radius 2.4 cm and slant height 6.3 cm.
The hemisphere has radius R cm.
The total surface area of the cone is equal to the total surface area of the hemisphere.

Calculate the value of R.

[The curved surface area, A, of a cone with radius r and slant height l is A = rrl .]
[The curved surface area, A, of a sphere with radius r is A = 4rr 2 .]

R = ................................................ [4]

© UCLES 2021 0580/41/M/J/21

 21

(b)

NOT TO
SCALE

16 cm

12 cm

7.6 cm 7.6 cm

The diagram shows a solid cone with radius 7.6 cm and height 16 cm.
A cut is made parallel to the base of the cone and the top section is removed.
The remaining solid has height 12 cm, as shown in the diagram.

Calculate the volume of the remaining solid.

[The volume, V, of a cone with radius r and height h is V = 13 rr 2 h .]

.......................................... cm3 [4]

© UCLES 2021 0580/41/M/J/21 [Turn over

 22

12 (a)
C

54°
x°
NOT TO
SCALE
5.3 cm
11 cm
G

6.9 cm

42°
A B

The diagram shows triangle ABC with point G inside.

CB = 11 cm, CG = 5.3 cm and BG = 6.9 cm.
Angle CAB = 42° and angle ACG = 54°.

(i) Calculate the value of x.

x = ................................................ [4]

(ii) Calculate AC.

AC = ........................................... cm [4]
© UCLES 2020 0580/43/M/J/20
 23

(b)

NOT TO
2.5 cm SCALE
15 cm

Water flows at a speed of 20 cm/s along a rectangular channel into a lake.

The width of the channel is 15 cm.
The depth of the water is 2.5 cm.

Calculate the amount of water that flows from the channel into the lake in 1 hour.
Give your answer in litres.

........................................ litres [4]

© UCLES 2020 0580/43/M/J/20 [Turn over

 24

12 (a)

C R

NOT TO
SCALE
A 8 cm B

P Q
12 cm

Triangle ABC is mathematically similar to triangle PQR.

The area of triangle ABC is 16 cm2.

(i) Calculate the area of triangle PQR.

.......................................... cm2 [2]

(ii) The triangles are the cross-sections of prisms which are also mathematically similar.
The volume of the smaller prism is 320 cm3.

Calculate the length of the larger prism.

............................................ cm [3]

© UCLES 2020 0580/42/M/J/20

 25

(b) A cylinder with radius 6 cm and height h cm has the same volume as a sphere with radius 4.5 cm.

Find the value of h.

4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3

h = ................................................ [3]

(c) A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius
r cm.

Find the value of r.

4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3

r = ................................................ [3]
7x
(d) A solid cylinder has radius x cm and height cm.
2
The surface area of a sphere with radius R cm is equal to the total surface area of the cylinder.

Find an expression for R in terms of x.

[The surface area, A, of a sphere with radius r is A = 4rr 2 .]

R = ................................................ [3]
© UCLES 2020 0580/42/M/J/20 [Turn over
 26

13

NOT TO
SCALE

8 cm 165° B
O

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]

(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 A = 4rr 2 .]

............................................ cm [4]

© UCLES 2020 0580/41/M/J/20

 27

(c)

NOT TO
h SCALE

r A
B

A cone is made from the sector by joining OA to OB.

(i) Calculate the radius, r, of the cone.

r = ........................................... cm [2]

(ii) Calculate the volume of the cone.

[The volume, V, of a cone with radius r and height h is V = 13 rr 2 h .]

.......................................... cm3 [4]

© UCLES 2020 0580/41/M/J/20 [Turn over