   SEARCH HOME Math Central Quandaries & Queries  Question from Lenny, a student: How do you find out the arc length of a sector 50 degrees wide?? We have two responses for you

Hi Lenny,

The length a of an arc is given by

a = r θ units

if r is the radius of the arc, θ is the angle measured in radians and the units are the same as the units used to measure the radius. There are π radians in 180o so if you measure the angle in degrees the length of the arc is given by

a = r π θ/180 units.

Thus if the angle is 50o then the length of the arc is

a = r π 50/180 = (5/18) π r units.

Penny

Hi Lenny.

The easiest way is this:

1. Find the length of the circumference. That's the arc all the way around the circle.

2. Multiply the circumference by the ratio of the degrees of the sector to the degrees of the whole circle (360).

For example, if I have a 45 degree sector of a circle whose radius is 5, then I know the whole circumference is 10pi. The ratio of the degrees of the sector to the whole circle is 45/360, which is just 1/8. So the arc length is 1/8 of 10pi.

Hope this helps,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.