CTK Insights

15 Feb

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) = 2n. 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 n > 1.

Here's an example of a solution for n = 7.

Divide a diameter of the circle into 7 equal parts. Now draw curves that consist of two semicircles: one above the diameter, the other below the diameter. The diameter of the left/upper semicircle takes up, say, k parts whilst the diameter of the right/lower part takes (7-k) parts.

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

π((m/2)² - (k/2)²) + π[((n - k)/2)² - ((n - m)/2)²]

This is equal to

π/8·2n(m - k).

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

πn/4.

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

π(n/2)² = π/4·n²

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.

9 Responses to “Dividing circular area into equal parts”

  1. 1
    Zac Says:

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

    Beautiful math indeed!

  2. 2
    Equal areas of a circle gives nice art - squareCircleZ Says:

    [...] Insights has an interesting post Dividing circular area into equal parts using a pair of compasses and a ruler, complete with [...]

  3. 3
    Bisecting Yin and Yang | CTK Insights Says:

    [...] 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 [...]

  4. 4
    Zadanie dopytu Says:

    Great post. Thank you for sharing this! :)

  5. 5
    Arjen Dijksman Says:

    Thank you for this nice circular area division art. It ia great source of inspiration for some developments on that theme.

  6. 6
    Varun Says:

    Thnx :)
    it hlpd a lot..
    keep up the gud work

  7. 7
    queensusu8 Says:

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

  8. 8
    shorturl.gamper.info Says:

    my site program pit 2013 (shorturl.gamper.info)

  9. 9
    AaBbCc Says:

    It helped but not what I needed

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