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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dividing circle into N equal parts


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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.

The derivation is simple and can be found in my blog. A GeoGebra illustration is available on a separate page.

References

  1. E. J. Barbeau, M. S. Klamkin, W. O. J. Moser, Five Hundred Mathematical Challenges, MAA, 1995, problem 88.

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