Solution
An Obtuse Viewing Angle
Show the problem again
Let A = (0, 0) and B = (1, 0) be the bottom corners of the unit square [0, 1] × [0, 1]. A point P is chosen uniformly at random inside the square. What is the probability that the angle APB is obtuse (greater than 90°)? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is π/8 ≈ 0.3927. By Thales' theorem, the angle APB is exactly 90° when P lies on the circle with diameter AB, larger than 90° inside that circle and smaller outside it. The circle has centre (1/2, 0) and radius 1/2; its part inside the square is the upper half-disk, with area (1/2)·π·(1/2)² = π/8. Since the square has area 1, the probability is π/8.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.