What is the probability that three points chosen at random inside a circle form an acute triangle?

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What's the probability that a random triangle is acute? (Gilbert Strang) from math

I answered a more general form of this problem here. Think of the integral as the sum of all the angles you can have, and the division as averaging it out. Just as $\frac{1}{n}\sum_{i=1}^n f(i)$ is an average, so is $\frac{1}{\pi}\int_0^\pi f(\theta)\mathrm d\theta$. So, in this case, you want to take the average of all the $\theta$ you can have, but there are an infinite amount of these, so how do you take an average? By this integral, of course!

Once you have an average $\theta$, you need to recognize that the length of the arc the third point can occupy in order to form the triangle is exactly $\theta$, the length of the minor arc between the first two points. The probability of landing in this arc is $\theta/2\pi$, and thus your integral.