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

**Problem**

Point *P* is randomly chosen inside the square *ABCD*. Lines *AP* and *PB* are then drawn. What is the probability that angle *APB* is obtuse?

**Solution**

Let θ = angle

*APB*

- If
*P*is within the white region, the angle is acute. See*P*_{1}. - If
*P*is on the red semi-circular curve, the angle is right. See*P*_{2}. - If
*P*is inside the semi-circular area, the angle is obtuse. See*P*_{3}.

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

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

$P = \pi / 8$ ← *answer*

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