Relationships Among Area, Perimeter and Volume

Perimeter measures the boundary; area measures the space inside it. Those are different jobs, so a fixed perimeter can box in very different areas depending on the shape. Take 40 meters of fencing. A 10 by 10 square uses all 40 meters and encloses 100 square meters. A 19 by 1 rectangle also uses 40 meters but encloses only 19 square meters. Same fence, wildly different yard.

Here's the mechanism. For a rectangle with perimeter P, the two side lengths add up to half of P, call it s. If the sides are s/2 + d and s/2 - d for some offset d, the area is (s/2)² - d^2. That's largest when d = 0, meaning both sides equal s/2 — a square — and it shrinks as d grows, heading toward zero as the rectangle gets thin and sliver-like. So area peaks at the square and falls away symmetrically on either side, which is why bigger perimeter never guarantees bigger area: shape matters as much as size.

This is why the standard pen problem always asks for the largest possible area, not just an area: with a fixed length of fencing, you're choosing a shape, and the square wins. When one side is a wall, you only fence three sides, so the trade-off shifts — the best shape is no longer a square but a rectangle twice as wide as it is deep, pushing the maximum area higher for the same fencing.