Question

We inscribed a square in a circle pf unit radius and shade the region between them. Them we inscribed another circle in the square and another square in the new circle and shaded the region between the new circle and the square. If the process is repeated infinitely many times, find the area of the shaded region.

Answer 1

if forms an $\infty$

GP series area

$=(\pi -2)+\frac{(\pi -2)}{\sqrt{2}}+\frac{(\pi -2)}{(\sqrt{2}{)}^2}+.......\infty$

$=\pi -2[1+\frac{1}{\sqrt{2}}+\frac{1}{(\sqrt{2}{)}^2}+......\infty ]$

$=(\pi -2)\frac{1}{(1-\frac{1}{\sqrt{2}})}$

$=\frac{(\pi -2)\sqrt{2}}{\sqrt{2}-1}$