Year 7 Volumes
Exercise 1 – Vertices, Edges and Faces
1.
[IMC 2015 Q7] A
tetrahedron is a solid
figure which has four
faces, all of which are
triangles. What is the product of the
number of edges and the number of
vertices of the tetrahedron?
2.
[IMC 2006 Q5] A solid ‘star’ shape is
created by gluing a square-based
pyramid, in which each edge is of
length 1 unit, precisely onto each face
of a cube of edge 1 unit. How many
faces does this ‘star’ have?
3.
[JMC 2004 Q8] A solid
square-based pyramid
has all of its corners cut
off, as shown. How
many edges does the resulting shape
have?
4. [JMO 2011 A5] The base of a pyramid
has 𝑛 edges. In terms of 𝑛, what is the
difference between the number of
edges of the pyramid and the number
of its faces?
5. [JMC 2002 Q19] The number of
diagonals of a regular polygon equals
twice the number of sides. How many
sides has the polygon?
6.
[IMC 2015 Q17] The football
shown is made by sewing
together 12 black pentagonal
panels and 20 white
hexagonal panels. There is a join
wherever two panels meet along an
edge.
How many joins are there?
7. [IMC 1999 Q14] Which of the
following statements is false?
A an octagon has twenty diagonals
B a hexagon has nine diagonals
C a hexagon has four more diagonals
than a pentagon
D a pentagon has the same number
of diagonals as it has sides
E a quadrilateral has twice as many
diagonals as it has sides

[IMC 1997 Q24] A regular
dodecahedron is a polyhedron with
twelve faces, each of which is a
regular pentagon. A
space diagonal of the
dodecahedron is a line
segment which joins
two vertices of the
dodecahedron which do not lie in the
same face. How many space diagonals
are there in the dodecahedron?
Exercise 2
1. Find the volume and surface area of
the following cuboid.
2. Find the volume and surface area of
the following cuboid.
3. The area of one side of a cube is
16m2. What is its volume?
www.drfrostmaths.com
4. Find the volume of the following solid.
5. A cuboid milk carton has a square
base of 8cm by 8cm and is 20cm
in height. I pour 400cm3 of milk
out into a glass. What is the
height of milk left in the carton?
6. Find the volume of the following solid.
7. [JMO 2006 A2] The
perimeter of this net of a
cube is 42cm. What is the
volume of the cube?
8. [JMO 2013 A3] The
solid shown is made by
gluing together four
1cm × 1cm × 1cm
cubes.
What is the total surface area of the
solid?
9. [JMC 2007 Q17] Just William’s cousin,
Sweet William, has a rectangular
block of fudge measuring 2 inches by
3 inches by 6 inches. He wants to cut
the block up into cubes whose side
lengths are whole numbers of inches.
What is the smallest number of cubes
he can obtain?
www.drfrostmaths.com
10. [JMO 1999 A6] A cube is made of
64 small cubes. Three holes are
made, with each hole
perpendicular to two faces and
passing right through the cube. The
shape and position of each hole is
shown in the diagram. How many
small cubes are in the remaining
solid?
11. [JMC 2008 Q14] A solid wooden cube
is painted blue on the outside. The
cube is then cut into eight smaller
cubes of equal size. What fraction of
the total surface area of these new
cubes is blue?
12. [JMC 2002 Q20] Sally has 72 small
wooden cubes, each measuring
1𝑐𝑚 × 1𝑐𝑚 × 1𝑐𝑚. She arranges
them all so that they form a cuboid.
Given that the perimeter of the base
of the cuboid is 16cm, what is its
height?
13. [IMC 2011 Q8] A square piece of card
has a square of side 2cm cut out from
each of its corners. The remaining
card is then folded along the dotted
lines shown to form an open box
whose total internal surface area is
180 cm2. What is the volume of the
open box in cm3?
14. [JMC 2011 Q20] One cube
has each of its faces covered
by one face of an identical
cube, making a solid as
shown. The volume of the solid is
875cm3. What, in cm2, is the surface
area of the solid?
15. [JMO 2005 A10] A closed rectangular
box is a double ‘cube’, in which the
top and bottom are squares, and the
height is twice the width. The greatest
distance between any two points of
this box is 9 cm. What is the total
surface area of the box?
16. IMC 1998 Q20] The total length of all
the edges of a cube is 𝐿 cm. If the
surface area of the cube has the same
numerical value 𝐿 cm2, what is its
volume in cm3?
A 1 B 𝐿
C 2
D 𝐿3 E 8
Exercise 3
1. A prism has a cross-sectional area of
7m2 and a length of 3m. What is its
volume?
2. Find the volume of this prism.
7. A square-based prism is 320cm3 in
volume. If its length is 20cm, what is
the side of the square?
8. [IMC 2010 Q17] Last year Gill’s
cylindrical 21st birthday cake wasn’t
big enough to feed all her friends. This
year she will double the radius and
triple the height. What will be the
ratio of the volume of this year’s
birthday cake to the volume of last
year’s cake?
A 12:1 B 7:1 C 6:1 D 4:1 E 3:1
 Find the volumes of the following
shapes, giving your answers in terms
of 𝑥 (and 𝜋 where applicable)
3. Find the volume of this prism.
4. Find the volume of this cylinder (to
3dp).
5. Find the volume of this prism.
6. RightUpYourAlleyTM are
manufacturing a new cylindrical toilet
roll with the pictured dimensions. As
usual the centre is hollow. Find the
volume of paper.
www.drfrostmaths.com
 [Edexcel] The pond is completely full
of water. Sumeet wants to empty the
pond so he can clean it. Sumeet uses
a pump to empty the pond. The
volume of water in the pond
decreases at a constant rate. The level
of the water in the pond goes down
by 20cm in the first 30 minutes. Work
out how much more time Sumeet has
to wait for the pump to empty the
pond completely.