jmc

Question

A circular sector has fixed perimeter, but the central angle

θ

and radius

r

are allowed to vary.

Which central angle

θ

maximizes the area of the circular sector? Give the answer in radians.

Note: The arc length of the sector is

r θ,

and the area of the sector is

\frac{r ^{2} θ}{2} .

Accepted Answer

Let P = 2r + r , the perimeter of the circular sector. By AM-GM, P = 2r + r 2 (2r)(r ) = 2 2r^2 .Then P^2 8r^2 , so r^2 2 P^216.Equality occurs when 2r = r , or = 2.