Perimeter is the length of the boundary or edge of a 2-D shape. Perimeter is measured in units of length such as millimetres (mm), centimetres (cm) or metres (m). The perimeter of a circle is called the circumference.

Area is a measure of surface. Area is measured in square units such as square millimetres (mm^{2}), square centimetres (cm^{2}) or square metres (m^{2}).

Volume is a measure of the space occupied by a liquid or a solid, or the space inside a container. Volume is measured in cubic units such as cubic millimetres (mm^{3}), cubic centimetres (cm^{3}) or cubic metres (m^{3}). A litre is equivalent to 1000 cm^{3} and 1 millilitre (ml) is equivalent to 1 cm^{3}.

Using fields and fences to explain perimeter and area helps clarify the concepts for children. The area is the size of the field and the perimeter is the amount of fencing. Pupils can draw pictures of fields on squared paper. They can count up the units of fencing around the edge.

Example:
To determine the area they can count the number of square units. To help with this, pupils could fill each square with a tree, therefore each tree represents one square unit.

To support the development of problem solving skills, get pupils to investigate the way a fixed area can produce a range of different perimeters or how a fixed perimeter can produce a range of different areas.

Additional User Example

A great investigation/problem solving for area and perimeter is the 'Farmer Brown' and his sheep pen problem:

Farmer Brown has 12 x 1m fence panels. How many different sheep pens can he make? What is the perimeter of each sheep pen? What is the area of each sheep pen?

This problem also lends itself well to differentiation - e.g. the sheep pen can be rectangular (or not). You can also vary the amount of perimeter fence and pupils can find out about different areas for different perimeters.

When pupils are investigating the problem it would be helpful for them to use practical equipment such as wooden stirring sticks to represent the 1 metre fence.

What this might look like in the classroom

Problem 1

The large rug in the twins’ bedroom was 6m long and 4m wide. The area of the floor of their bedroom was 80m^{2}. How much of the floor was not covered by the rug?

Solution:

First work out the area of the rug: 6 m × 4 m = 24 m^{2}. Next take that away from the area of the floor: 80 m^{2} − 24 m^{2} = 56 m^{2}
Which strategies did you use to solve this problem?

Problem 2

Sami was asked to work out how much fencing would be needed to surround a garden with an area 144m^{2}. He hasn’t been told the perimeter of the garden. What are the options for the amount of fencing needed?

Solution:

You would need to work out the perimeter of the garden. There are various options, all of which will be factor pairs of 144: 1 m × 144 m, 2 m × 72 m, 3 m × 48 m, 4 m × 3 m, 6 m × 24 m, 8 m × 18 m, 12 m × 12 m. Remembering that you will need to add two of each measurement the perimeters will be: 290 m, 148 m, 102 m, 80 m, 60 m, 52 m, 48 m.

In which areas of maths could you incorporate this kind of problem?

Activity 3:

Match the cards to their correct place on the grid:

Solution:

Note that some of the cards go in more than one place, and misconceptions are addressed to encourage exploration of the distinction between area and perimeter; i.e. a child might get perimeter and area confused and place a card incorrectly, but then find that they can’t place all of the remaining cards. The always, sometimes, never true possibilities allow pupils to explore general statements and discuss responses using appropriate mathematical vocabulary.

Problem 4

I filled a container with water. The container was 10 cm, wide, 10 cm deep and 12 cm high. What is the volume of water that I needed to fill it?

Solution:

The volume is the amount of space an object takes. Imagine the water in the container and the space it would take. You would need to multiply the width by the depth by the height: 10 × 10 × 120. An obvious way to answer this would be to multiply 10 by 10 to give 100 and then multiply 120 by 100 giving 12000 ml^{3} or 12 l^{3}.

Taking this mathematics further

Real life contexts It is important to discuss where area and perimeter are relevant to real life so that children can see a purpose for these concepts, for example, decorating using wall paper, carpeting a room.
Setting up problems for example designing a bedroom will help. The children could begin by designing the shape and size of their room scaled down to a suitable unit, measuring it out on paper and then working out floor coverings, appropriate sizes of bed, desk, cupboards, wardrobe etc.
You could also look at floor plans of houses, from the internet or estate agents and work out areas of rooms from their dimensions.

Making connections

Children need the basics of various mathematical concepts to be able to answer problems involving area and perimeter. Firstly, they need to have had practical experience of measuring length with increasing accuracy. They need to understand the concepts of perimeter as the distance around the outside of a shape and the area as the amount within the perimeter. They would do this for squares and oblongs, initially on squared paper, by counting sides of squares for the perimeter and whole squares for area. They would then measure with a ruler the width and length of the shape and add 2 × width to 2 × length or 4 × one side if a square for perimeter. They would calculate the area by multiplying the width times the length.

When confident with these methods they would begin to use the formula 2(l + w) for perimeter and l × w for area.

It is essential that children use units correctly. Take the opportunity to address any confusion and avoid the temptation to accept the correct numerical value of the answer, while glossing over the wrong choice of units. Incorrect choice of units signifies a lack of understanding about the distinction between perimeter, area (and volume). The power of the unit (being either 1, 2 or 3) links neatly to the dimensions of the area in question − explore the reasons for this with children.

KS2 children do not usually encounter volume, but should they wish to, it would be best if they begin by building shapes practically, perhaps using interlocking cubes, and counting the number used.

Related information and resources from the NCETM

Learning maths outside the classroom: Following the DCSF publication of its manifesto 'Learning Outside the Classroom' in 2006, Year 3 from Corbridge Primary School began a project that involved badgers and area and perimeter. This article gives an account of their 'School Grounds' project.

Related information and resources from other sites

Area and perimeter: This is an interactive resource to help children understand about area and perimeter from Woodlands Junior School. This site also offers other activities in the area of mathematics.