Unit 16 Section 4 : Formulae for circumference and area

There are formulae to find the size of the area (A) and circumference (C) of a circle.
To find either of these we need to use the size of the radius (r) or diameter (D).

Radius and Diameter
The length of the radius goes from the edge to the centre of the circle, and the diameter goes all the
way from edge to edge through the centre, so the diameter is exactly twice the length of the radius.

If we know the radius then we double it to get the diameter, and if we know the diameter then we halve it to get the radius.

Diameter and Circumference
The circumference of a circle is just over three times the diameter, so to calculate the circumference
we need to multiply the diameter by a value which is a bit bigger than 3. In fact, the value we have to
multiply by is called pi and is represented by the greek letter pi which looks like this:

The value of pi is 3.14159265... and the decimal part of the number carries on for ever without recurring.
Normally we use the button marked on our calculator to solve problems involving pi, but if we do
need to work by hand or we only have a basic calculator then we tend to use 3.14 as an approximation.

If we know the circle diameter D then we multiply it by (pi) to get the circumference C.
This is normally written as a formula:

C = D

We can reverse the process too: if we know the circumference then we can divide it by to find the diameter.

Note that if we know the radius we would multiply it by 2 to get the diameter, and then multiply by pi to get the circumference.
This can be seen in the other formula for circumference:

C = 2r

Radius and Area
The radius and area of a circle are also linked by this number which is roughly 3.14.

The formula to find the area A using the radius r is:

A = r²

It is very important to realise that the r² part of the calculation is done before you multiply by . This is because
BiDMAS tells us that indices (like squaring a number) are calculated before multiplications. So, if we know the
circle radius r we can square it and then multiply by to find the area A.

Reversing this process is slightly trickier: to go from the area back to the circumference we need to divide by pi and
then square-root the result. It is important to get these two operations the right way round.

Summary
The diagram below summarises the operations needed to do calculations involving the measurements in a circle:

Practice Questions

Practice Question 1
A circle has radius 6 cm.

(a) Calculate its area, accurate to 1 decimal place.

(b) Calculate its circumference, accurate to 1 decimal place.

Practice Question 2
A circle has diameter 7 m.

(a) Calculate its circumference, accurate to 1 decimal place.

(b) Calculate its area, accurate to 1 decimal place.

Practice Question 3
The circumference of a circle is 18.2 cm.

Calculate the length of the diameter of the circle, accurate to 1 decimal place.

Practice Question 4
The area of a circle is 22.8 m².

Calculate the length of the radius of the circle, accurate to 1 decimal place.

Exercises

GIVE YOUR ANSWERS TO 1 DECIMAL PLACE IN ALL THE QUESTIONS BELOW USE THE BUTTON ON YOUR CALCULATOR OR 3.14 FOR PI

Question 5
Look at the circle below:

What is the circle's area? cm²

Question 6
A circle is cut out of a rectangular piece of card as shown below:

(a) Calculate the area of the rectangle. cm² (b) Calculate the area of the circle. cm² (c) What is the area of the remaining card after the circle is cut out? cm²

Question 7
Look at the semicircle shown below.

Calculate the perimeter of the semicircle. cm