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