Discussion
The shape of a curve in the plane with fixed perimeter that encloses maximum area is :
Answer: D
The curve is a circle.
This is the isoperimetric problem: among all closed plane curves of a given length \(L\), the one enclosing the greatest area is the circle. The isoperimetric inequality states \(4\pi A \le L^{2}\), with equality only for a circle. In the calculus of variations it is solved by maximising the area integral with the length as a constraint (a Lagrange multiplier), and the Euler equation says the curvature is constant.
For perimeter 4, the square has area 1, a 1.5 by 0.5 rectangle has 0.75, while the circle has area \(4/\pi \approx 1.27\). An ellipse with unequal axes also encloses less than the circle of the same perimeter.
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