Kaleidoscope Of Circles

Geometry Level 5

A circle \(C\) of \(r=10\) is divided into \(n\) equal sectors (the image above shows the case for \(n=8\)). In each of them, the largest circle touches the arc and two surrounding radii, the second circle is tangent to the first circle and two radii, the third tangent to the second, and this goes on forever.

If the total area of these small circles in the original circle \(C\) is \({ A }_{ n }\), find \[\displaystyle \pi { r }^{ 2 }-\lim _{ n\to\infty }{ { A }_{ n } } \; . \]

Give your answer to 3 decimal places.

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