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Sounds perfect Wahhhh, I don’t wanna
mirkokosmos
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shonk
shonk

In which I answer the question “What percentage of triangles are acute?” and babble about polymers. See more explanation here, and check out SciState.

strangewastelandpersona

I don’t think the question is well defined. I considered the space of triangles a different way, and found that 50% are acute.

Every triangle (that isn’t colinear, but let’s ignore those) can be inscribed uniquely in a circle, so up to scaling we can describe the triangles as the sets of three points on the boundary of a given circle. A triangle is non-acute iff the three points lie in the same semicircle, so once the first two points are chosen, making angle θ with the centre, say, then the arc of points that would make an acute triangle also has angle θ. If you assume the points on the circle are chosen with uniform probability, once you integrate everything out (assuming I’ve done it correctly) you get a probability of ½ that a random triangle is acute.

Edit: Shonk is correct that my method’s probability is ¼, not ½. I clearly need to revise integration…

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