What’s The Probability The Angle Is Acute?

A point P is chosen at random inside a square QRST. What is the probability angle RPQ is acute?

Give the problem a go before jumping in for the solution!

Solution

From knowledge of circle geometry, we know that if the angle RPQ is 90 degrees, then P lies on a semicircle of diameter RQ.

Notice any angle formed within the semicircle is greater than 90 degrees. And any angled formed outside the semicircle is acute.

We now let the side length of the square be x.

The area of the semicircle is

Therefore, the probability is

Thank you for reading - Barry