## Area of Circles 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. 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

\[\frac{10}{3^k}\]

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

\[A= \frac{a}{1-r}=\frac{\frac{100 \pi }{9}}{1-1/9}=\frac{800 \pi}{9}\]

.