In my probability book there was a question of finding the probability of choosing a random point from circle inscribed in circle, which I found out to be $pi$/$4$. After solving this question I thought of its reverse scenerio of the same problem in which square is inside in a circle and I have to find out probability of choosing random point from square inscribed in a circle, this time.
My Calculations so far:\
Let $E$ be the event of choosing point from square.
For square inscribed in a circle with radius $r$ and Area = $\pi r^2$, we have diagonal of square given by: D = $2r$
I can't seem to figure out how do find the area of circle now from this point? because that's the only information I need to find the desired result. i.e: P(E) = Area of square/Area of circle
Best Answer
The diagonal of the square is in fact $2r$. Using Pythagoras, the length of the square, $x$, is given by $$x^2+x^2=4r^2\implies 2x^2=4r^2\implies x^2=2r^2\implies x=r\sqrt2$$ Hence the area of the square is $(r\sqrt 2)^2=2r^2$
Divide the area of the square by the area of the circle and there's your answer.