A square is inscribed in a circle with radius 'r'. What is the probability that a randomly selected point within the circle is not within the square?

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Solution

The correct option is C

Let the square be ABCD.

Length of the diagonal of the square ABCD = (r + r) = 2r

As $Δ$ BCD is right-angled at C, applying Pythagoras theorem:

$(B C)^{2} + (C D)^{2} = (B D)^{2}$

$\Rightarrow a^{2} + a^{2} = (2 r)^{2}$

$\Rightarrow 2 a^{2} = 4 r^{2}$ $\to a = \sqrt{2} r$

Area (sq ABCD) = $2 r^{2}$
Area (circle) = $π r^{2}$

Area (circle) - Area (sq ABCD) = $r^{2} (π - 2)$

So, P (that a randomly selected point within the circle is not within the circle)

$= \frac{(A r e a (c i r c l e) - A r e a (s q A B C D))}{A r e a (c i r c l e)}$ $= \frac{(r^{2} (π - 2))}{π \times r^{2}}$

$= \frac{(π - 2)}{π}$

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