Answer:
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.
