4.7 E) Area of Sector
You may be asked in the exam to work out the area or perimeter of a single sector. Let’s have a few examples.
What is the area of the following sector? Give your answer to 2 decimal places.
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°. We are able to see what proportion of a full circle we have by dividing the angle for the sector by 360°. We divide by 360° because there are 360° in a full circle.
We find out the area of the sector by multiplying the proportion of the full circle that we have by the area of the whole circle. This give us the calculation that is shown below.
The area of the sector is 19.63 cm2.
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.
What is the exact area of the following sector?
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 angle 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.
The area of the sector is 7.5π m2.
Working Backwards
Sometimes it will be the case that you are given the area of a sector and asked to find out the radius or the angle of the sector. We answer questions like this by letting the equation for the area of a sector equal the area of the sector. We then sub the radius of the sector (or angle of the sector) into the formula and solve to find the angle of the sector (or radius of the sector).
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. We are told that the area is 27π cm2 and the angle of the sector is 120°.
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.
End Note
It may be the case that we are given the area of a sector and asked to work out the angle of the sector. We answer questions like this in essentially the same way as example 3. The only difference is that we will be solving the equation for the angle rather than for r (the radius). There are a few questions that ask you to find out the angle of the sector in the quiz.