Question

A square, circle, regular hexagon and regular octagon all have the same perimeter \(P\). Which one has the maximum area?

• A. Square
• B. Circle
• C. Hexagon
• D. Octagon

Hint:

With equal perimeter, increasing the number of sides of a regular polygon increases area; the circle (limit as \(n\to\infty\)) maximizes it.

Answer 1

The Correct Option is B

Key fact (isoperimetric principle). Among all plane figures with a fixed perimeter, the circle encloses the greatest area. Quantitative comparison (regular \(n\)-gon formula).
For a regular \(n\)-gon with perimeter \(P\), side \(s=P/n\), apothem \(a=\dfrac{s}{2}\cot\left(\dfrac{\pi}{n}\right)\), and area \[ A_n=\frac12\cdot P \cdot a = \frac{P^2}{4n}\cot\!\left(\frac{\pi}{n}\right). \] Compute for the given shapes: \[ \begin{aligned} A_{\text{square}}&=\frac{P^2}{16},
A_{\text{hex}}&=\frac{P^2}{24}\cot\!\left(\frac{\pi}{6}\right) =\frac{P^2}{24}\,(\sqrt{3})\approx \frac{P^2}{13.856},
A_{\text{oct}}&=\frac{P^2}{32}\cot\!\left(\frac{\pi}{8}\right) =\frac{P^2}{32}\,(1+\sqrt{2})\approx \frac{P^2}{13.257},
A_{\text{circle}}&=\frac{P^2}{4\pi}\approx \frac{P^2}{12.566}. \end{aligned} \] Since \(\dfrac{1}{12.566}>\dfrac{1}{13.257}>\dfrac{1}{13.856}>\dfrac{1}{16}\), the circle’s area is the largest, followed by the octagon, hexagon, and square. \[ \boxed{\text{Circle}} \]