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Step-by-step explanation:

Lets start by fixing the amount of fence available for both structures, call this P for perimeter.

If the side length of the triangle is s, then P=3s, and s=P/3.

Using Pythagorean theorem and Area=(1/2)base*height, you can see that Area of Triangle =\sqrt(3) * (s^2)/4

Substituting P/3 for s, yields P^2 * (\sqrt(3)/36).

Now for a square with side length x, and Area x^2, P=4x, so we can substitute and get Area of Square = P^2 (1/16)

Now we compare the formulas for Area in terms of P, and note that 1/16 is larger than (\sqrt(3)/36), thus the square encloses a bigger area than the triangle.

In general the more sides we have the more area we enclose, and a circle (which kind of has infinitely many "sides") encloses the maximal area for a fixed perimeter.

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step-by-step explanation: