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

For any ##theta##, the length of the arc is given by the formula (if you work in radians, which you should: 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##