How to work out the arc length and perimeter of a sector (part of the circumference of a circle)

Arc Length Of A Sector Video


If you take a part of the circumference of a circle then the distance along this arc is called the arc length. So since the arc length is part of the circumference of a circle, then the arc length of the sector can be found by using the following formula:

l = (x/360) × π × d

Where l is the arc length, x is the angle inside the sector and d is the diameter of the sector.

So make sure that you use the diameter of the circle if you are calculating the arc length of a sector.

Example 1

Work out the arc length of this sector:


The angle inside the sector is 41⁰, so x = 41⁰.

Now the diameter of the whole circle is 24cm (as the 12cm is the radius), so d = 24cm.

All you need to do now is substitute these values into the above formula so you can find the arc length.

l = (x/360) × π × d

l = (41/360) × π × 24

l = 8.6 cm rounded to 1 decimal place.

Example 2

Work out the arc length of this sector:

The angle inside the sector is 134⁰, so x = 134⁰.

Now the diameter of the whole circle is 18cm (as the 9cm is the radius), so d = 18cm.

All you need to do now is substitute these values into the above formula so you can find the arc length.

l = (x/360) × π × d

l = (134/360) × π × 18

l = 21.0 cm rounded to 1 decimal place.

Example 3

A sheet of metal is shaped in the form of a sector with a radius of 6cm and an angle of 308⁰. Work out the perimeter of the sheet of metal.

First you need to calculate the arc length of the sector. The diameter of the sector is 12cm (d = 12) and the angle of the sector is 308⁰ (x = 308⁰)

l = (x/360) × π × d

l = (308/360) × π × 12

l = 32.3 cm rounded to 1 decimal place.

Now, the question asks for the perimeter of the sector (the distance around the whole of the shape)

So you need to add on two radiuses onto the arc length:

P = 32.3 + 6 + 6 = 44.3cm

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1 comment

MazioCreate 5 years ago

Thanks again catman3000 for making what appears to be a difficult equation, simple. The grey matter has be suitably stirred to meet the day with that formula. I must have used it at some time but haven't needed to go down that path for decades. It is all so easy with a calculator. How did I ever carry out complicated maths at school? No calculator and no idea how to use an abacus to decimals, I must have gone through a lot of paper and pens.

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