A circle is not a polygon with infinitely many sides, geometry insists
Add sides to a regular polygon and it looks ever more like a circle. A shape with 10,000 equal sides has interior angles of 179.964 degrees. Yet the angle can never reach exactly 180, because at that point the edges would flatten into a straight line, so the familiar claim that circles are infinite polygons fails.
A regular polygon has every side the same length and every angle equal. That simple demand forces a great deal of order. All its corners sit on one surrounding circle, and a second circle fits snugly inside, touching each side at its midpoint. A regular shape with n sides can be turned n ways onto itself and flipped across n mirror lines, which together form what mathematicians call the dihedral group. When n is even, half those mirror lines run corner to corner and half run from one side's middle to the opposite side's; when n is odd, each runs from a corner to the middle of the far side.
Angles follow a neat rule: the exterior angles of any convex regular polygon always add up to one full turn, 360 degrees, so each interior angle climbs toward, but never touches, a straight line as sides multiply. Keep the edge length fixed instead of the size and the limiting shape is not a circle at all but an apeirogon, effectively an endless line.
Some surprising identities hold. Put a regular polygon inside a circle of radius one, and multiply the distances from one corner to every other corner: the product equals the number of sides. From any point inside, the perpendicular distances to all the sides sum to the same total, n times the distance from centre to side, extending Viviani's theorem for triangles.
Construction has limits. With only compass and straightedge, a regular n-gon is possible exactly when every odd prime dividing n is a different Fermat prime, primes of a special form tied to powers of two. Origami, by contrast, can reach additional shapes. Diagonals grow quickly too, from none in a triangle to nine in a hexagon.
Source: Regular polygon
