The perimeter of a sector is the length of the outside of the sectors. A sector is shown below:

There are three different sides that we need to consider when we are working out the perimeter of a sector. These are the two radii and the arc length. The formula for working out the perimeter of a sector is given below:

*Where r is the radius and d is the diameter.*

The first part of the formula is for the two radii and the second part of the formula is to work out the arc length. The two radii of a sector do not change if the angle of the sector increases or decreases. But, the arc length does change if the angle of the sector increases or decreases; if the angle of the sector was to increase, the arc length would increase and so would the perimeter of the sector (and if the angle was to decrease, the arc length would decrease and so would the perimeter of the sector).

Let’s have a few examples.

**Example 1**

What is the perimeter of the sector below? Give your answer to one decimal place.

We are given the angle and the radius in the diagram. We are able to work out the diameter by doubling the radius that we are given; the diameter of the sector is 14 cm (7 x 2). We now have everything that we need for the formula, so we just sub the numbers in for their respective places.

The perimeter of the sector is 17.7 cm to one decimal place.

**Example 2**

The exact perimeter of the sector below is (16 + 13π) m. What is the angle of the sector?

We are going to be using the generic formula to find the angle of the sector. The generic formula is shown below:

We are now going to sub in the values that we are given into this formula. We are given the perimeter of the sector [(16 + 13π) m], the radius of the sector (8 m) and we can work out the diameter by doubling the radius (the diameter is 16 m [8 x 2]). Let’s now sub these values in.

The above equation at the moment looks quite complex. Therefore, the first step should be to tidy up the equation, which we can do by completing some of the calculations.

We can see that there is a 16 on both sides of the equation. We can get rid of the 16 on both sides by taking 16 from both sides of the equation.

There is a π on both sides of the equation and we can get rid of this π by dividing both sides of the equation by π.

The next to find what the angle is for this equation is to deal with the “x 16” at the end of the right side of the equation. We are able to get rid of this by doing the opposite; we divide both sides of the equation by 16 – it is easier if we leave the left side of the equation as a fraction.

Currently we have a fraction on the right side of the equation. We are able to get rid of fractions by multiplying both sides of the equation by the denominator of the fraction. This means that we need to multiply both sides of the equation by 360.

The angle of the sector is 292.5°.