Maximum Area With a Fixed Length of Fencing: Why a Square Wins and How a Wall Changes It

The Algebra

Let the rectangle have width w and length L, with perimeter 2w + 2L = 100, so L = 50 − w. The area is A = w × L = w(50 − w) = 50w − w². This is a downward-opening parabola, with its maximum at the vertex, w = −b ÷ 2a = 50 ÷ 2 = 25. Then L = 25, and the area is 625 square feet. Without calculus, you can also see that A = 625 − (w − 25)², which is largest when w = 25 and decreases as w moves away from 25 in either direction.

Using a Wall or River as One Side

If one side of the enclosure is an existing wall, you only need fence for three sides. Let the two sides perpendicular to the wall each have length w, and the side parallel to the wall have length L. Fencing: 2w + L = 100, so L = 100 − 2w. Area = w(100 − 2w) = 100w − 2w². The maximum is at w = 100 ÷ 4 = 25, giving L = 50 and an area of 1,250 square feet, exactly double the area of the free-standing square. The optimal shape has the side along the wall twice as long as each of the two sides. Using a wall (or a fence line you already have) is a very efficient way to double the area.

Other Shapes and the Circle

A regular polygon with more sides encloses more area for the same perimeter. A square with a perimeter of 100 feet has an area of 625. A regular hexagon has 100 ÷ 6 = 16.67 foot sides and an area of 2.598 × 16.67² = 722. A circle with circumference 100 feet has radius 15.9 feet and area π × 15.9² = 796 square feet, the maximum possible. So a circular pen uses fence most efficiently, though it is harder to build and fit into a yard, and gates and corner posts add complexity. A circle uses about 11% less fence than a square for the same area.

In practice, rectangles are easier to build, to fit against property lines, and to use for garden beds.

Practical Considerations

The mathematical optimum may not be practical. A long thin garden bed is easier to reach and tend than a square one. A dog run might need to be long for exercise. Gate openings take up fence length. Posts take up some of the length. Use the optimization as a baseline and compare it to your needs. If you have a fixed area to enclose and want to minimize fence, a square (or a rectangle close to square) uses the least material: for 625 square feet, a square needs 100 feet of fence, whereas a 10 by 62.5 foot rectangle needs 145 feet, 45% more.

The General Lesson

Optimization problems appear throughout business and engineering: maximizing profit, minimizing cost, or finding the best dimensions of a container. The recipe is the same: express what you want to optimize in terms of one variable using the constraint, then find the maximum or minimum, using algebra, a graph, or calculus. Always check that the answer makes physical sense.