Circular sector of minimum perimeter

An optimisation problem with a constraint: the area of a circular sector is fixed and one seeks the radius that minimises its perimeter.

Example — Circular sector of fixed area with minimum perimeter

A circular sector has area $A=100$. Find the radius $r$ that minimises the perimeter.

Solution

Setting up. The area of the sector is $A = \frac{1}{2}r^2\theta = 100$, hence $\theta = 200/r^2$. The perimeter is the sum of the two radii and the arc ($r\theta$): $P = 2r + r\theta = 2r + \frac{200}{r}.$

Derivative: $P'(r) = 2 - \dfrac{200}{r^2} = 0 \iff r^2 = 100 \iff r = 10$.

$P_{\min} = 2\cdot 10 + 200/10 = 20+20 = \boxed{40}$.

Links

Topics: Derivatives
Concepts: Objective function · Maximum and minimum
Methods: Maximum minimum via derivative
Skills: Differentiating · Modelling