40 MENSURATION
CHECK YOU CAN: BY THE END OF THIS CHAPTER YOU WILL BE ABLE TO:
l use metric units for l use metric units of mass, length, area, volume and capacity in
length, area, volume practical situations and convert quantities into larger or smaller units
and capacity l carry out calculations involving the perimeter and area of a rectangle,
l understand the triangle, parallelogram and trapezium
geometric terms used l carry out calculations involving the circumference and area of a circle
with shapes l carry out calculations involving arc length and sector area as fractions
l rearrange formulas and of the circumference and area of a circle
solve equations l carry out calculations and solve problems involving the surface area
l round numbers to and volume of a: cuboid, prism, cylinder, sphere, pyramid and cone
a given number of l carry out calculations and solve problems involving perimeters and
decimal places or areas of: compound shapes, parts of shapes
significant figures. l carry out calculations and solve problems involving surface areas and
volumes of: compound solids, parts of solids.
Forcalculating the surface
area of a pyramid and a
cone, you will also need
to be able to: The perimeter of a 2-D shape
l use Pythagoras’
theorem. The perimeter of a shape is the distance all the way around the edge of the shape.
For calculating the Since the perimeter of a shape is a length, you must use units such as
volume and surface centimetres (cm), metres (m) or kilometres (km).
area of more complex
compound shapes, you
will also need to be Example 40.1
able to: Question Solution
l use Pythagoras’ Find the perimeter of this shape. Perimeter = 0.6 + 1.4 + 1 + 1.5 = 4.5 m
theorem
1.5 m
l calculate lengths in
similar figures
l find the area of a non-
Note
1m Although you don’t
right-angled triangle
using the formula have to give the units in
1
bc sinA. 0.6 m your working, you must
2 remember to give the
units with your answer.
1.4 m
Sometimes not all of the lengths of the shape are given in the diagram.
Before trying to find the perimeter, work out all the lengths.
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The perimeter of a 2-D shape
Example 40.2
Question
Find the perimeter of this shape made from rectangles.
9 cm
These two must add
to give 9 cm
The missing length
5 cm is therefore
9 – 5 = 4 cm.
10 cm
7 cm
These two must add
to make 10 cm
The missing length
is therefore
10 – 7 = 3 cm.
Solution
Perimeter = 9 + 10 + 4 + 7 + 5 + 3 = 38 cm
Exercise 40.1
1 Find the perimeter of each of these rectangles.
a b
3.5 cm 14 cm
1.5 cm 5 cm
2 Copy each of these diagrams, where the shapes are made from rectangles.
Find any missing lengths and mark them on your diagram.
Find the perimeter of each shape.
a b
20 cm
12 cm
7 cm 10 cm
16 cm
6 cm
7 cm
20 cm 8 cm
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40 Mensuration
c d
200 m 2 cm
40 m
3.4 cm
3.1 cm 8.2 cm
130 m
2 cm
3.4 cm
80 m
3 Measure each of these shapes accurately.
Work out the perimeter of each shape.
a b c
4 The perimeter of a rectangle is 26 cm.
Two sides are 8 cm long.
How long are the other sides?
5 A square has a perimeter of 120 cm.
How long is each side?
6 A rectangle has a perimeter of 60 cm.
The lengths of its sides are in whole centimetres.
What are the possible sizes of the rectangle?
The area of a rectangle
The area of a two-dimensional shape is the amount of flat space inside
the shape.
Note
Whenever you are giving For rectangles of any size,
an answer for an area,
make sure you include the area of a rectangle = length × width
units.
If the lengths are in
centimetres, the area will
be in cm2.
width
If the lengths are in
metres, the area will be
in m2.
length
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The area of a triangle
Exercise 40.2
1 A rectangle measures 4.7 cm by 3.6 cm.
Find its area.
2 A square has sides 2.6 m long.
Find its area.
3 A rectangle measures 3.62 cm by 4.15 cm.
Find its area.
4 A rectangular pond measures 4.5 m by 8 m.
Find its area.
5 An airport is built on a rectangular piece of land 1.8 km long
and 1.3 km wide.
a Calculate the area of the land.
b Find the length of fencing needed to go round the perimeter of
the airport.
6 The perimeter of a square is 28 cm.
Calculate its area.
7 A rectangle has an area of 240 cm 2.
One of its sides is 16 cm long.
Calculate the length of the other side of the rectangle.
8 A rectangular lawn measures 24 m by 18.5 m.
a Work out the area of the lawn.
Lawn weedkiller is spread on the lawn.
50 g of weedkiller is needed for every square metre.
b How many kilograms of weedkiller are needed to treat the whole lawn?
c Weedkiller is sold in 2.5 kg boxes.
How many boxes are needed?
The area of a triangle
Triangle PQR has base length b and perpendicular height h. X P Y
Rectangle XYQR has the same base length and height.
The blue area is the same size as the pink area. h
1
Area of the rectangle is bh, so the area of the triangle is 2
bh.
R Q
1 1 b
Area of a triangle = × base × perpendicular height or A =
2 2
bh
Example 40.3
Question
a Find the area of this triangle. 6 cm
8 cm
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40 Mensuration
Note b The area of this triangle is 20 cm2.
Find the perpendicular height of the triangle.
Always use the
perpendicular height of h cm
Solution
the triangle, never the
a Area = 1 bh
slant height. 2
8 cm
In this triangle, =1×8×6
2
area = 1 × 6 × 3 = 9 cm2.
2 = 24 cm2
b Area = 1bh
2
5 cm 20 = 1 × 8 × h
2
3 cm
20 = 4h
h=5
6 cm
So the height is 5 cm.
Remember that the units of area are always square units, such as square
centimetres or square metres, written cm 2 or m2.
When using the formula, you can use any of the sides of the triangle as the
base, provided you use the perpendicular height that goes with it.
Exercise 40.3
1 Find the area of each of these triangles.
a b c
8m
4 cm
10 cm
6 cm 5m 7 cm
d e f
7m 6.2 cm 4.5 m
9m 9.8 cm 5.6 m
g h i
5m
5.2 cm 8m
6m
3m 6.3 cm
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The area of a parallelogram
2 Find the area of each of these triangles.
a b c
9 cm
10 m 9m
8m 10 cm 11 m
d e f
5m 5.8 m
4.3 cm 3.4 m
7m 6.4 cm
g h i
7.5 cm
8.3 m
5.1 m
3 cm
3.2 m
4.6 m
3 In triangle ABC, AB = 6 cm, BC = 8 cm and AC = 10 cm. Angle ABC = 90°.
a Find the area of the triangle. b Find the perpendicular
A height BD.
D
10 cm
6 cm
B 8 cm C
The area of a parallelogram
A parallelogram may be cut up and rearranged to form a rectangle or two
congruent triangles.
1
Area of a rectangle = base × height Area of each triangle = 2 × base × height
Both these ways of splitting a parallelogram show how to find its area.
Area of a parallelogram = base × height
Note
Make sure you use the
perpendicular height
height and not the sloping edge
when finding the area of
a parallelogram.
base
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