# Area and Perimeter

 From: Tom Stone Date: 8 April 1999 Subject: Relating Area and Perimeter On our GNVQ engineering course we were set this problem: Show that the area of a circle is proportional to the square of its perimeter. Can you help me on this? To find the area or perimeter of a circle you need to know the radius, don't you? So how can you do this?

### Maths Help suggests:

Yes, the formula for the area and the perimeter (or circumference) of a circle are both functions of the radius. To get a relationship between the area and the perimeter, you need to eliminate the radius. Do it like this:

Make r the subject of the first:

and substitute this into the second:

So, to 4 decimal places, we have the result
A = 0.0796 P2
which demonstrates that the Area is proportional to the Perimeter squared. The constant of proportionality is 0.0796

Remember what is meant by "Y is proportional to X".
This means that you can show that Y = k.X
where k is a constant.
We have shown that Area (A) is proportional to the Perimeter squared (P2)
with the constant k being 0.0796 for circles.

It is a useful fact that the area of any regular two-dimensional shape is proportional to its perimeter. Let's run through a similar proof for squares.