Why is a manhole cover round? Because a square manhole cover could fall into the hole it's supposed to cover. But a round manhole cover cannot, as long as the hole is ever so slightly smaller. What property of a circle makes this true?
Is it because the circle has constant diameter? If so, we're going to have to be able to clearly abstract the idea of diameter from the circle to other shapes. So, what does a diameter mean for other shapes? What is the diameter of a square of side 1, for instance?
For any 2D geometric shape, we can define a diameter by imagining that we try to fit the shape entirely between two straight lines. Some orientations of the shape will allow the two lines to come close, and some far. To check all the possible orientations, we could imagine driving a nail through one point of the shape, turning it around that point, and measuring the widths that the shape takes up. During this process, we may get many values; we are particularly interested in the maximum and minimum values that turn up, which we will call the maximum diameter and minimum diameter of the shape.
It should be clear that the circle has constant diameter, that is, equal maximum and minimum diameters. The square of side 1 has maximum diameter sqrt(2) and minimum diameter 1.
Now, the question is,
Is the circle the only shape with constant diameter?
I give up, show me the solution.