Take a piece of string. As everyone knows from experience, the largest area that can be enclosed by a string of a given length is a disk. But what happens if the string is stretched between three pegs? In other words, which triangle with a given perimeter has the greatest area?
Intuition suggests that symmetry makes the equilateral triangle the answer. Proving this seems far more difficult. And yet Heron's formula provides an easy way through!
Round and round…
Let a triangle have sides a, b and c, and let p denote its semiperimeter and S its area. The arithmetic–geometric mean inequality states that, for any three given positive numbers x, y and z,
(xyz)13≤x+y+z3\left(xyz \right)^ {\frac{1}{3}} \le \dfrac{x+y+z}{3}