If a square is drawn inside a circle with radius r, what is the probability that a randomly chosen point within the circle falls inside the square?

Example Answer

I would clarify whether the square is inscribed, meaning all four vertices lie on the circle, and whether the point is chosen uniformly over the circle's area. Under those assumptions, the square's diagonal is the diameter, 2r. Its side is r√2, so its area is 2r². Dividing by the circle's area, πr², gives 2/π, or about 63.7%.

A square merely drawn somewhere inside the circle can be smaller. If its side length is s and it is fully contained in the circle, the uniform-area probability is s²/(πr²). Without its size, the question does not determine a single probability.

Why this works

Area ratios give probabilities only for uniform sampling over area. Sampling the radius uniformly from zero to r and the angle uniformly does not produce a uniform-area point; it places too much density near the center.

Follow-up to prepare for

To simulate uniform-area sampling, use radius r × sqrt(U) and angle 2πV, where U and V are independent uniform random variables on [0, 1].