Question

(i) A circle is inscribed in a square. A point inside the square is randomly selected. What is the probability that the point is inside the circle as well?

(ii) If, instead, the square was inscribed in the circle, and a point inside the circle was randomly selected, what is the probability that it is inside the square?

Answer 1

(i) If the circle has a radius of r, then the square will have a side of 2r.
Area of square = 4$r^2$
Area of circle = $\pi$$r^2$
Probability that the point lies in circle = $\frac{\pi r^2}{4r^2}$ = $\frac{\pi }{4}$

(ii) If the circle has a radius of r, then the square will have a diagonal of 2r and side $\sqrt{2}$r
Area of square = 2$r^2$
Area of circle = $\pi$$r^2$
Probability that the point lies in square = $\frac{2r^2}{\pi r^2}$= $\frac{2}{\pi }$