Mensuration

Worksheet 3

Composite Solids

1. W20/qp22/q4(a)(i)(ii)

The diagram shows a solid formed by joining a cylinder to a hemisphere. The

diameter of the cylinder is 9cm and its height is 16cm.

i. The volume of the hemisphere is equal to the volume of the cylinder.

Show that the radius of the hemisphere is 7.86cm, correct to 2
decimal places. [4]
ii. Calculate the total surface area of the solid. [3]
2. W19/qp21/q4

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 8cm. The height of the cylinder is
95cm and the height of the cone is 15cm.

a. Calculate the volume of the gate post. [3]

b. Show that the total curved surface area of the gate post is 2580𝑐𝑚2 ,
correct to 3 significant figures. [5]
c. A geometrically similar gate post has a total height of 150cm.
Calculate the total curved surface area of this gate post. [2]
3. W18/qp21/q9(a)

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 3cm.
The total height of the lamp is 24cm.

i. Show that the volume of lamp A is 650𝑐𝑚3 , correct to 3 significant

figures. [4]
ii. Calculate the curved surface area of lamp A. [3]
iii. Lamp B is mathematically similar to lamp A. The volume of lamp B is
450 𝑐𝑚3 . Calculate the total height of lamp B. [2]
4. W17/qp21/q8

The diagram shows solid A which is made from a hemisphere joined to a cone of
equal radius. The hemisphere and the cone each have radius 6cm. The total height of
the solid is 18cm.

a. Show that the slant height, 𝑥 cm, of the cone is 13.4cm, correct to 1
decimal place. [2]
b. Calculate the total surface area of solid A. [3]
c. Calculate the volume of solid A. [3]
d. Solid A is one of a set of three geometrically similar solids, A, B and C.
The ratio of the heights of solid A : solid B : solid C is 2 : 6 : 1.
i. Calculate the surface area of solid B correct to 3 significant
figures. [2]
ii. Calculate the volume of solid C correct to 3 significant figures.
[2]
5. S15/qp21/q4

A solid is formed by joining a cone of radius 4.5cm and height 7.6cm to a hemisphere
of radius 4.5 cm as shown.

a. Calculate the area of the circle where they are joined. [2]
b. Calculate the total volume of the solid. [2]
c. Another solid of the same type is made by joining a cone of radius
5cm and height h cm to a hemisphere of radius 5cm. The cone and
hemisphere have equal volumes. Calculate the height of the cone.[2]
6. W12/qp22/q7

A cylindrical, open container has a diameter of 21 cm and height of 8 cm.

a.
i. Calculate the total external surface area of this container. [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. [2]
b. A circular top that can hold 4 hemispherical bowls can be placed on the
container.

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. [1]
ii. Calculate the total surface area of the top of the container, including the
inside curved surface area of each bowl. [3]
iii. With the top and the 4 bowls in place, calculate the volume of water
required to fill the container. [3]
7. S11/qp22/q11

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. [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]
ii. Find the circumference of the base of the cone. [2]
iii. The curved surface area of the cone can be made into the
shape of a sector of a circle with angle θº. Show that θ is 200,
correct to the nearest integer. [2]
iv. Hence, or otherwise, find the total surface area of the solid.
[3]
8. W22/qp41/q1 (0580)
a. Calculate the volume of
i. a solid cylinder with radius 6cm and height 14cm, [2]
ii. a solid hemisphere with radius 6cm. [2]
b. The cylinder and hemisphere in part (a) are joined to form the solid in the
diagram. The solid is made of steel and 1 𝑐𝑚3 of steel has a mass of 7.85g.
i. Show that 1 𝑐𝑚3 of steel has a mass of 0.00785kg. [1]

ii. Calculate the total mass of the solid. [2]

c. 2000 𝑐𝑚3 of iron is melted down and some of it is used to make 50 spheres
with radius 2 cm.
i. Calculate the percentage of iron that is left over. [3]
ii. The iron left over is then made into a cube
Calculate the length of an edge of the cube. [1]
d. A solid cone has radius 3𝑅𝑐𝑚 and slant height 9𝑅𝑐𝑚. A solid cylinder has
radius 𝑥 𝑐𝑚 and height 7𝑥 𝑐𝑚. The total surface area of the cone is equal to
the total surface area of the cylinder. Given that 𝑅 = 𝑘𝑥,
find the value of k. [4]
9. W21/qp42/q7(a) (0580)

The diagram shows a container for storing grain. The container is made from a
hemisphere, a cylinder and a cone, each with radius 2m. The height of the cylinder is
5.2m and the height of the cone is hm.
i. Calculate the volume of the hemisphere. Give your answer as a
multiple of 𝜋. [2]

88𝜋
ii. The total volume of the container is 𝑚3 . Calculate the value of
3

h. [4]
iii. The container is full of grain. Grain is removed from the container at
a rate of 35000kg per hour. 1𝑚3 of grain has a mass of 620kg.
Calculate the time taken to empty the container. Give your answer in
hours and minutes. [3]
10. S18/qp41/q6 (0580)

A solid hemisphere has volume 230𝑐𝑚3 .

a. Calculate the radius of the hemisphere. [3]

b. A solid cylinder with radius 1.6cm is attached to the hemisphere to
make a toy. The total volume of the toy is 300𝑐𝑚3 .

i. Calculate the height of the cylinder. [3]

ii. A mathematically similar toy has volume 19200𝑐𝑚3 . Calculate
the radius of the cylinder for this toy. [3]
 Answer Key

1. W20/qp22/q4(a)(i)(ii)

2. W19/qp21/q4
3. W18/qp21/q9(a)

4. W17/qp21/q8

5. S15/qp21/q4
6. W12/qp22/q7

7. S11/qp22/q11
8. W22/qp41/q1 (0580)
9. W21/qp42/q7(a) (0580)

10. S18/qp41/q6 (0580)