1. S8 Perimeter, area and volume
KS3 Mathematics
The aim of this unit is to teach pupils to:
Deduce and use formulae to calculate lengths, perimeters, areas and volumes in 2-D and 3-D shapes
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
S8 Perimeter, area and volume

2. S8 Perimeter, area and volume
Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle

3. What is the perimeter of this shape?
To find the perimeter of a shape we add together the length of all the sides.
What is the perimeter of this shape?
Starting point
1 cm 3
Perimeter =
= 12 cm 2 3
Ask pupils if they know how many dimensions measurements of perimeter have. Establish that they only have one dimension, length, even though the measurement is used for two-dimensional shapes.
Tell pupils that when finding the perimeter of a shape with many sides it is a good idea to mark on a starting point and then work from there adding up the lengths of all the sides. 1 1 2

4. Perimeter
Sometime we are not given the lengths of all the sides. We have to work them out from the information we are given.
9 cm
12 – 5
a =
= 7 cm
5 cm
b =
9 – 4
= 5 cm
12 cm
4 cm
Discuss how to work out the missing sides of this shape.
The side marked a cm plus the 5 cm side must be equal to 12 cm, a is therefore 7 cm.
The side marked b cm plus the 4 cm side must be equal to 9 cm, b is therefore 5 cm.
a cm
7 cm
P =
= 42 cm
5 cm
b cm

5. S8 Perimeter, area and volume
Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle

6. Area
The area of a shape is a measure of how much surface the shape takes up.
For example, which of these rugs covers a larger surface?
Rug B
Rug A
Rug C
Discuss how we can compare the area of the rugs by counting the squares that make up each pattern.
Conclude that Rug B covers the largest surface.

7. Area of a rectangle Area is measured in square units.
For example, we can use mm2, cm2, m2 or km2.
The 2 tells us that there are two dimensions, length and width.
We can find the area of a rectangle by multiplying the length and the width of the rectangle together.
length, l
width, w
This formula should be revision from key stage 2 work.
Area of a rectangle
= length × width
= lw

8. Area of a rectangle What is the area of this rectangle? 4 cm 8 cm
The length and the width of the rectangle can be modified to make the arithmetic more challenging.
Different units could also be used to stress that units must be the same before they are substituted into a formula.
Area of a rectangle = lw
= 8 cm × 4 cm
= 32 cm2

9. Area of shapes made from rectangles
How can we find the area of this shape?
We can think of this shape as being made up of two rectangles.
7 m
Either like this … A
10 m
15 m
… or like this.
8 m
Label the rectangles A and B.
Discuss ways to divide this composite shape into rectangles.
A third possibility not shown on this slide would be to take the square of area 15 m × 15 m and to subtract the area of the rectangle 10 m × 8 m.
This gives us 225 m2 – 80 m2 = 145 m2. B
5 m
Area A = 10 × 7 = 70 m2
15 m
Area B = 5 × 15 = 75 m2
Total area = = 145 m2

10. S8 Perimeter, area and volume
Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle

11. Making cuboids The following cuboid is made out of interlocking cubes.
How many cubes does it contain?

12. Making cuboids
We can work this out by dividing the cuboid into layers.
The number of cubes in each layer can be found by multiplying the number of cubes along the length by the number of cubes along the width.
3 × 4 = 12 cubes in each layer
There are three layers altogether so the total number of cubes in the cuboid = 3 × 12 = 36 cubes

13. Making cuboids
The amount of space that a three-dimensional object takes up is called its volume.
Volume is measured in cubic units.
For example, we can use mm3, cm3, m3 or km3.
The 3 tells us that there are three dimensions, length, width and height.
Link:
S7 Measures – units of volume and capacity
Liquid volume or capacity is measured in ml, l, pints or gallons.

14. Volume of a cuboid
We can find the volume of a cuboid by multiplying the area of the base by the height.
The area of the base
= length × width
So,
height, h
Volume of a cuboid
= length × width × height
= lwh
length, l
width, w

15. Volume of a cuboid What is the volume of this cuboid? Volume of cuboid
= length × width × height
5 cm
= 5 × 8 × 13
8 cm
13 cm
= 520 cm3