An interesting related problem is Bertrand's paradox[0]. You have an equilateral triangle inscribed in a circle. You pick a random chord from the circle (a line that goes from one edge of the circle to another). What's the probability the chord is longer than an edge of the triangle?

As it turns out, there isn't a well-defined solution. That's the paradox. The issue comes from "picking a random chord". There are different ways that you can pick a random chord which lead to different solutions. One approach to picking a random chord is to pick two random points on the circle and draw the chord between them. This gives you a probability of 1/3. Another approach is to pick a random direction and radius, then draw a chord at the end of the radius perpendicular to the radius. This approach gives you a probability of 1/2.

One interesting aspect is the second approach is the only approach where if you were to draw a circle within the larger circle, the distribution of the chords in the inner circle will match the distribution of chords in the outer circle.

[0] https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)