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

1 Answer
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#