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My teacher gave us an interesting problem today. Consider a circle of radius $1$, choose three points on that circle at random and make a triangle connecting the three. On average what will the area of the triangle be?

One kid eventually got it and he told him to do it with $4$ points on a sphere making a tetrahedron and to find the average volume.

This was in a homeroom class, where the students in the class had him the previous year in AP Calc, so I'm not sure what the tags should be or what math is used.

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    $\begingroup$ This problem is known as Circle triangle picking and the expecte area of triangle is $\frac{3}{2\pi}$. The 3d version is known as Sphere tetrahedron picking and the expected volume of tetrahedron is $\frac{4\pi}{105}$. In general, these sort of stuff should fall under the topic "geometric probability". $\endgroup$
    – achille hui
    Commented Apr 29, 2017 at 4:20
  • $\begingroup$ @achillehui Thank you! $\endgroup$
    – Jacob Claassen
    Commented Apr 29, 2017 at 4:24
  • $\begingroup$ @achillehui: Doesn't one encounter the same type of problem as in Bertrand's paradox? $\endgroup$
    – Alex M.
    Commented Apr 29, 2017 at 17:32
  • $\begingroup$ One quick simplification is that if you fix one point and choose the other two at random, the answer is the same because of symmetry. $\endgroup$
    – Michael Hardy
    Commented Apr 29, 2017 at 19:45
  • $\begingroup$ @AlexM. Bertrand paradox talks about the difficulty in specifying this type of problem unambiguously, not the problems themselves. $\endgroup$
    – achille hui
    Commented Apr 30, 2017 at 8:33

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