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If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first reaction was "0" or "undefined". But my son wrote "$\infty$" which I think is a reasonable answer.
The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality). The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry, and it has rotational symmetry around the centre for every angle.
6 Σεπ 2019 · If we’re adamant that a circle really is a polygon with infinitely many sides, then the question presents itself: what are the sides? And surely the only plausible answer [1] is: the individual points of the circle.
So, a circle technically has zero sides.
Here is my approach to a proof that a circle has infinitely many sides: We consider an equilateral n-gon. The n vertices are equidistant from the center. If n is infinity, there is an infinite number of vertices that are equidistant from the center. That is the definition of a circle.
If you think of the circle as a disk then it it has an up-side and a down-side. If you think of it as a curve, then it has an inside and an outside. If you think of it as the limit of an n-sided regular polygon, then one can justify the answer that the circle has infinitely many infinitesimal sides.
