# How to divide a circle into 9 rings / 1 inner circle with the same area?

The objective is to divide a circle of any size into 10 equal areas where 1 is a smaller inner circle and 9 are rings.

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Can you plz illustrate with a figure? –  Swapnanil Saha Jan 4 '13 at 13:50
So what you want is a set of circles with areas $0.1\cdot A$, $0.2\cdot A$, $\ldots, 1.0\cdot A$ ($A$ is the area of the original circle), and stack them on top of eachother, right? What are your limitations? (With straightedge and compass this is hard, with calculator, ruler and compass it's easy.) –  Arthur Jan 4 '13 at 13:52
@Arthur: it only requires taking square roots of rational numbers. With straightedge and compass it is not that hard. –  Willie Wong Jan 4 '13 at 14:08
This is what I want to achieve: imgur.com/NlJdi –  user55170 Jan 4 '13 at 15:15

Let the radius of the outer circle be $R$. Let the radii of the $10$ circles be $r_1, r_2, \ldots r_{10}$ As the area of a circle is is $A=\pi r^2$, you want $\frac {\pi R^2}{10}=\pi (r_i^2-r_{i-1}^2)$, with $r_{10}=R$ and $r_0=0$. This gives $r_i=R\sqrt {\frac {i}{10}}$

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Thank you for your awnser that seems to be what I need, do you know a way to apply it automatically or can anyone give me a graphical representation with it applied? –  user55170 Jan 4 '13 at 15:16
@user55170: I don't know what you mean by apply it automatically. You could just take graph paper, let $R=1$, and plot circles $\sqrt {0.1} \approx 0.3162, \sqrt {0.2} \approx 0.447$ and so on to get a look. –  Ross Millikan Jan 4 '13 at 15:47
Im trying to create this form with Adobe Illustrator and trying to figure out how to go from the formula to a complete drawing. I've added this image above: : imgur.com/NlJdi –  user55170 Jan 4 '13 at 15:57
@user55170: you have kept the radii as $0.1, 0.2. 0.3$ and so on. The area then is ($\pi$ times) $0.1^2, (0.2^2-0.1^2), (0.3^2-0.2^2)$ and so on. You need the center much larger, as in my last comment, if you want the areas to be the same. Right now your outer ring has $19$ times the area of the center circle. –  Ross Millikan Jan 4 '13 at 16:22
@user55170: Note my edit to the answer-I pulled the $R$ out of the square root at the end, fixing a typo. The units didn't work before. –  Ross Millikan Jan 4 '13 at 16:23

Just for a lark (and to confirm my comment), below is (almost) all the steps required to build the picture using compass and straightedge. (Created using kseg.)

The steps skipped: starting from the original circle, take its radius, and divide that by 10. (Arbitrary integer division is available in compass and straightedge.) Starting from just two points separated that length apart, the image below can be constructed in 68 steps. (Those two initial points are in green.)

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Btw: the construction is based on the square root spiral. –  Willie Wong Jan 4 '13 at 17:25
Thank you for creating this, its easier now that I can see how when properly done it looks like. –  user55170 Jan 4 '13 at 17:32