That's the question I asked my sons today. It's an interesting problem but to make it a little more interesting and apropos given we're in Australia, I asked them: "If you were a crocodile farmer and only had a 100 foot long fence, what shape would you arrange it in to have the most space for your reptiles?"
I got many different answers: Square! Pentagon! Circle! Oval! Most people assumed square was the right answer but didn't all want to pick the same shape.
So we decided to start with a more general square: a rectangle. Thanks to a length of rope (representing our 100 foot fence) we were able to prototype various areas. A square seemed best. Was that so?
We turned to Mathematica (I just bought the newly released version 8
, love it
) for confirmation:
The maximum area is achieved when a side is equal to 25 feet. In other words: a square gives us the max area of 625 ft^2.
What about other shapes? Mathematica to the rescue!
A triangle makes poor use of fencing, the square is an improvement but clearly the more sides our fence has, the greater the area... The optimal shape must be a circle!
Though they saw that the circular field's area was greater than the square field's, this still didn't hit home until I plotted both.