How do you calculate the length of an arc and the area of a sector?

1 Answer
Nico Ekkart
Dec 19, 2014

For any theta, the length of the arc is given by the formula (if you work in radians, which you should:
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The area of the sector is given by the formula (theta r^2)/2

Why is this?
If you remember, the formula for the perimeter of a circle is 2pir.
In radians, a full circle is 2pi. So if the angle theta = 2pi, than the length of the arc (perimeter) = 2pir. If we now replace 2pi by theta, we get the formula S = rtheta

If you remember, the formula for the area of a circle is pir^2.
If the angle theta = 2pi, than the length of the sector is equal to the area of a circle = pir^2. We've said that theta = 2pi, so that means that pi = theta/2.
If we now replace pi by theta/2, we get the formula for the area of a sector: theta/2r^2

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