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​4.7 E) Area of Sector
A sector is an area that is enclosed by 2 radii. The circle below shows two sectors. 
When we have a whole circle and it is split into two sectors, the larger one of the sectors is known as the major sector and the smaller sector is known as the minor sector.
 
You may be asked in the exam to work out the area and perimeter of a single sector. Let’s have a few examples. 
Example 1 
What is the area of the following sector? Give your answer to 2 decimal places.
From a previous section, we learnt that we are able to work the area of a circle out by using the following formula (click here to be taken through to the section about working out the area of a circle):
r in the above formula stands for the radius.
 
The formula above is the formula for a full circle. However, our sector is not a full circle. We are told that the angle is a right angle, which means that it is 90°. A full circle is 360°. We are able to see what proportion of a full circle we have by dividing the angle for the sector by 360°. 
Therefore, we have one quarter of a full circle.
 
The next step is to work out for the whole circle. We will then multiply the value that we obtain by the proportion of the circle that the sector is (we will multiply the area by one quarter). 
The radius of our sector is 5 cm, so we can sub r as 5 into the equation above.

The area of the sector is 19.63 cm2.

Generical Formula 
The generic formula for working out the area of a sector is given below:
​It is worth writing this formula down on a revision card. 
Example 2 
What is the exact area of the following sector?
We are asked to find the exact area of the sector, which means that we need to leave our answer in terms of π.
 
For the sector, we are given the angle that is not included in the sector; this angle is 60°. We need to find out what the area is for the sector. We know that a full circle is 360° and this means that we can work out the angle of the sector by taking 60° away from 360°. This means that the angle of the sector is 300°. As we have the angle of the sector (300°) and the length of the radius (3 m), we are now able to work out what the area of the sector is by subbing these values into the generic formula.
​We want to obtain the exact value, so we just need to make sure that when we get the answer from a calculator, we leave the answer in terms of π. 

The area of the sector is 7.5π m2.

Working Backwards

Example 3

The sector below has an exact area of 27π cm2. What is the radius of the sector?

We are given the area and the angle of the sector. From this information, we will be able to work out what the radius of the sector is. We will do this by using the generic formula for the area of a sector to find the value for r. The generic formula for the area of a sector is:

Let’s now sub the values that we are given into this formula.

When we look at this formula, we will notice that there is a π on both sides of the equation and this means that we can divide both sides of the equation by π, which eliminates π from the equation. The equation becomes:

Also, 120/360 simplifies down to 1/3.

The next step is to make r2 the subject. In order to do this, we need to get rid of the fraction, which we are able to do by multiplying both sides by the denominator of the fraction; we therefore multiply both sides by 3.

We want r and not r2, so we need to square root both sides.

Therefore, the radius of the above sector is 9 cm.