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How did Dido get the idea with the half-circle? Could there be a way to get even more land with the same bull's skin?
To prove that Dido did get the biggest piece of land possible, we'll look at the relationship of different shapes with the same perimeter to their areas. To make it easier we'll first forget about the trick Dido used with the seashore and pretend we have to put the bull's hide all the way around the land.
To start off, let's make some different rectangles and find
out how big the area is. A rectangle where two sides
are 4 and two sides 10 units long, like the one on the right, will do fine
for a start. You can easily
calculate the area of that square, which is 40.
Now, to see how the relations are when you change the shape of the rectangle,
we'll look at another rectangle where the sides also add up to 28. Let's try
this one where two sides are 3 and two 11 units long.
When we calculate the area we get
33. Oops, that's
a smaller area than we had before.
