Divide a Circle into N Parts of Equal Area
Is it possible to divide a circle into N parts of equal area the Euclidean way: with straightedge and compass?
You'd be within your rights to have doubts. An immediate solution that comes to mind is to divide the circle into N equal arcs by vertices of a regular N-gon. However, as it turns out, not all N-gons are constructible. For example, the regular heptagon -- the polygon with 7 vertices -- is not.
But this is just a slight hindrance: the problem has an elegant solution for any N! Seek and you will find.
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Each curve inside the big circle consists of two semicircles. The centers of all semicircles involved are collinear. The diameter of the big circle is divided into N equal parts.
Curiously, all the curves inside the circle are of the same length as the big semicircle. Any pair of adjacent ones bounds a region of the same area. Put another way, all regions have the same perimeter, which is equal to the circumference of the big circle. They also have equal areas.
- E. J. Barbeau, M. S. Klamkin, W. O. J. Moser, Five Hundred Mathematical Challenges, MAA, 1995, problem 88.