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Right again! The more sides a polygon has, the bigger its area. I'm sure you also noticed that the polygons with forty or more sides look more like a circle than a polygon. So it probably doesn't surprise you that the more sides an equilateral polygon has, the closer its area will get to the one of a circle.
So now we have just shown that the circle is the shape which has the biggest area for a given perimeter. First we saw that for any shape with x number of sides, the shape which has the biggest area is the one where all the sides and all the angles are the same (and for a given perimeter there is only one of these shapes). Then we found out that the more sides our equalateral polygon has, the bigger the area will be and the closer it will be to a circle which gives the biggest area. Therefore the shape with the biggest area for a given circumferenc is the circle. That's exactly what Dido found out too.
But for Dido, a circle was not good enough. She wanted a
bigger piece of land. So she decided to use the seashore as one side.
Since you already know that without a seashore the circle
is the best shape, you'll easily see that the half circle is the best
shape if you have one given side. We won't prove that now. But
did Dido get a much bigger area because she used the seashore? The answer
is definitely yes. By using the seashore as one of the sides, Dido got
exactly twice as much land as she would have, had she just laid out a
circle.
