### Dividing circular area into equal parts

It's not simple and sometimes impossible to divide a circle into n equal parts. That is, if the word "circle" signifies the border of a circular region, then, as was shown by the great Gauss, the construction with compass and ruler is only possible when Euler function φ is a power of 2: φ(n) = 2^{n}. This happens exactly when n is the product of Fermat primes and a power of 2. Here are a few such numbers 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24 ... In particular Euclidian construction of a regular heptagon, i.e. a polygon with 7 sides is impossible.

However, if the question is of dividing the area of a circle, not the circumference, the problem is solvable in an elegant way that works for any

Remarkably, the length of each of the curves equals the length of the original semicircle. More Importantly, each of the regions bounded by two successive curves is exactly 1/7 of the given circle.

Let's check this in a general case of n parts. Take two curves: one on, say, k and n-k parts, the other on m and n-m parts, m >k. Assuming each part equals 2 so that k, n-k, m, and n-m are the radii of the four semicircles, the area enclosed by the two curves is half the sum

This is equal to

In case m = k + 1, i.e., when the region at hand is bounded by two successive curves the area is

But, in our notations, the radius of the big original circle is n/2 and the area is

which is exactly n times the area between two successive curves.

For an interactive construction with a Java applet, visit Divide a Circle into N Parts of Equal Area.

Thanks for this - I haven't come across it before.

Beautiful math indeed!

April 30th, 2008 at 11:09 pm[...] Insights has an interesting post Dividing circular area into equal parts using a pair of compasses and a ruler, complete with [...]

May 1st, 2008 at 12:22 am[...] one of the very first posts on the blog I wrote about a problem of dividing a circle into parts of equal area. More recently I [...]

May 24th, 2009 at 9:33 amGreat post. Thank you for sharing this!

March 14th, 2010 at 1:48 pmThank you for this nice circular area division art. It ia great source of inspiration for some developments on that theme.

March 14th, 2010 at 4:34 pmThnx

December 12th, 2010 at 7:32 amit hlpd a lot..

keep up the gud work

i love dividing but i don"t know how to do it

March 10th, 2011 at 7:09 pmmy site program pit 2013 (shorturl.gamper.info)

July 23rd, 2013 at 9:02 pmIt helped but not what I needed

April 8th, 2014 at 9:04 am