Circles - Arcs Sectors

## Circles - Arcs Sectors

An arc is a fraction of the circumference of a circle.

The amount of the circumference that is the arc is given by the number of degrees at the centre.

The circumference of the whole circle is given by 2pir, and the angle at the centre is 360º.

If the angle of the arc is 83º, then the fraction is frac(83)(360) of a whole circle:

The arc length is therefore frac(83)(360) xx 2pir.

Note that if the perimeter of the sector is required, then include the two radii: frac(83)(360) xx 2pir + 2r.

The area of a whole circle is pir^2. The area of a sector uses the same fraction: frac(83)(360) xx pir^2.

## Example 1

What is the area of the sector, shown below?

 Area of a circle A = pir^2 Area of this sector A = frac(70)(360) xx pir^2 Substitute A = frac(70)(360) xx pi xx 12^2 = 87.96

 Perimeter of a circle P = 2pir Perimeter of a sector P = frac(70)(360) xx 2pir + 2r Substitute P = frac(70)(360) xx 2pi(12) + 2(12) = 14.65 + 24 = 38.66 round to 1dp = 38.7