Question

A circle is inscribed in a regular hexagon,a regular hexagon is inscribed in that circle and so on...
What's the sum of the areas of all those circles?

Answer 1

The inradius, r, and circumradius, R, of a hexagon can be related as \[{R \over r} = {2 \over \sqrt{3}} = \left({R \over r}\right)^2 = \left({4 \over 3}\right)\]Therefore, each inscribed circle has a radius which is sqrt(3)/2 times the radius of the next largest circle. We can write the area of the largest inscribed circle as \[A_1 = \pi r^2\]The area of the second circle becomes\[A_2 = \pi \left({3 \over 4}\right)r^2\]where r is the radius of the largest inscribed circle.

Writing the summation of this\[A_{\text{total}} = A_1 + \pi \sum_{i=1}^{n} \left(3 \over 4\right)^i r^2 \]

Answer 2

I'll refer you here for more information on hexagons: http://mathworld.wolfram.com/Hexagon.html

and here fore more information on summations: http://mathworld.wolfram.com/Sum.html

Answer 3

thanks!