Area of Circles Filling a Sector

A circle radius 10cm is cut into six equal sectors. Inside each sector, circles of decreasing radius are drawn to fill the sector as shown.

circle filling a sector

The radius of the largest circle is the solution to  
\[r+\frac{r}{sin \pi /6}=3r=10 \rightarrow r= \frac{10}{3} cm\]
The smaller circles extend out to  
\[10-2r=10- 2 \frac{10}{3}=\frac{10}{3} cm \]
  from the centre.

circle filling a sector

Repeating the above process for the next largest circle gives us a radius for this circle of  
\[\frac{1}{3} \frac{10}{3} = \frac{10}{9}\]
In general the radius of the kth largest circle will be  
  and the area of the circle is  
\[\pi (\frac{10}{3^k})^2 = \frac{100 \pi }{9^k}\]
The areas of the circles form a geometric sequence with first term  
\[a = \frac{100 \pi }{9}\]
  and common ratio  
\[r = \frac{1}{9}\]
The area of all the circles is  
\[A= \frac{a}{1-r}=\frac{\frac{100 \pi }{9}}{1-1/9}=\frac{800 \pi}{9}\]

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