Perimeter is the length of the boundary or edge of a 2D 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 measured in square units such as square millimetres (mm^{2}), square centimetres (cm^{2}) or square metres (m^{2}).

Volume is measured in cubic units such as cubic millimetres (mm^{3}), cubic centimetres (cm^{3}) or cubic metres (m^{3})

What this might look like in the classroom

Activity 1:

Match the cards to their correct place on the grid:

A circle has a perimeter

Value of area = value of perimeter

Area means the same as perimeter

Solution 1:

A circle has a perimeter

Value of area = value of perimeter

Area means the same as perimeter

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 pupil might get perimeter and area confused and place a card incorrectly, but then find that they can’t place all of the remaining cards. Encourage pupils to reason why mistakes might occur.

(Based on an idea from www.kangaroomaths.com)

Problem 2:

A 3 cm by 6 cm rectangle is an ‘equable shape’ as its perimeter has the same numerical value as its area (18 cm and 18 cm^{2} respectively). Investigate and find other examples of equable shapes. Can you find a square which is an equable shape?

Solution 2:

There are several shapes that have an area and perimeter with the same value; e.g.

A rectangle with sides 10 cm and 2.5 cm

A right-angled triangle with sides 5 cm, 12 cm and 13 cm

A square of side 4 cm

This activity should be used to promote clarity between the terms area and perimeter. It highlights the fact that area and perimeter can have the same value, but only rarely.

Problem 3:

Following on from Problem 2, a 3D shape is said to be equable if its surface area is numerically equivalent to its volume. What equable 3D shapes can you find?

Solution 3:

There are several shapes that have a surface area and volume with the same value; e.g. a cube of side 6 cm

Taking this mathematics further

Research fractals. While they are generally considered to be self-repeating shapes, the original term actually refers to a dimension that is not an integer. This was first discussed by the mathematician Benoit Mandlebrot in a publication called “How long is the coastline of Britain?” (1967)
The most well-known example of a fractal is probably the Koch Snowflake. This shape has a finite perimeter and an infinite perimeter and results in a dimension of about 1.26. It is however, fairly straightforward to describe.
Other related objects include the Sierpinkski Sponge, the Mandlebrot Set and the Julia Set. While the technical mathematics is quite complex, its results are likely to amaze, and there are a remarkable number of applications including medicine, seismology and computer-game design.

Making connections

It is essential that pupils 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 learners.

Care is needed when dealing with 3D shapes and distinguishing between surface area and volume – which is perhaps more understandable given that a 3D shape had a 2D and 3D measure associated with it.

The more abstract idea of establishing the dimensions of a formula and whether it could represent an area, volume or perimeter, is rooted in effective exploration of the nature of possible formulae at this early stage.