Probability that a Point Inside a Square Will Subtend an Obtuse Angle to Adjacent Corners of the Square

Let θ = angle APB
 

 

• If P is within the white region, the angle is acute. See P₁.
• If P is on the red semi-circular curve, the angle is right. See P₂.
• If P is inside the semi-circular area, the angle is obtuse. See P₃.

 

$P = \dfrac{A_\text{semi-circle}}{A_\text{square}}$

$P = \dfrac{\frac{1}{2}\pi (0.5^2)}{1^2}$

$P = \pi / 8$   ←   answer