I know that for a fixed perimeter, the shape that encloses the greatest area
is a circle, and the rectangle that encloses the greatest area is a square.
Sadly, I don't know how to prove it to you without Calculus.
If you'll take my assertion that the greatest rectangle is a square, and accept
it on faith, then you should use your 160-yd of fence to enclose a square with
40-yd sides. The area inside it is (40 x 40) = 1600 square yards.
Here are some other choices.
Each one has the same perimeter ... 160 yards.
This table kind of suggests to you that a square is the rectangle
with the greatest area. (But it doesn't prove it.)
Length Width Area
35-yd 45-yd 1,575 square yards
30 50 1,500
25 55 1,375
20 60 1,200
15 65 975
10 70 700
5 75 375
3 77 231
2 78 156
1 79 79
2-ft 79-yd 1-ft 52.89
1-ft 79-yd 2-ft 26.56
1-inch 79-yd 35-in 2.19 square yards
With the same 160-yd of fence, you could have squeezed in some more
area by setting the fence down in a circle with circumference = 160-yd.
The area inside the circle would be
Area = (pi) (radius)² = (circumference)² / (4 pi) = 2,037.2 square yards
The area of a circle is always (4 / pi) times the area of the square with the
same perimeter. That's about 27.3% more area than the square.
