The sides of a rectangle are chosen at random, each less than a given length a, all such lengths being equally likely. The chance that the diagonal is less than $\alpha $ is

Let the rectangle sides be x and y, where 0 < x < a and 0 < y < a, giving a sample space of area a squared. We need x squared plus y squared to be less than alpha squared, which defines a quarter-circle of radius alpha in the first quadrant with an area of pi times alpha squared divided by 4. The probability is the ratio of this quarter-circle area to a squared, which simplifies to pi divided by 4.

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