Single Choice

We inscribe a square in a circle of unit radius and shade the region between them then we inscribe another circle in the square and another square in the new circle and shade is region between the new circle and this square if the process is repeated infinitely many times , the area of the shaded region

A$$ 2 \pi $$
B$$ 3( \pi - 2)$$
C$$ 2 ( \pi - 2 ) $$
DNone of these
Correct Answer

Solution

$$ A_ 1 = \pi (1)^2 - ( \sqrt {2} )^2 $$
$$ = \pi - 2 $$
$$ \sqrt {2} a = \frac {1}{ \sqrt {2} } $$
$$ a = \frac {1}{2} $$
$$ A_2 = \pi ( \frac {1}{ \sqrt {2} })^2 - ( \frac {1}{2})^2 $$
$$ = \frac { \pi}{2} - \frac {1}{4} $$
$$ = \frac { 2 R - 1}{4 } $$

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