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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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2 Answers

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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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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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