On the one hand, I think that by symmetry the average side of a triangle with given perimeter $p$ is $\frac{p}{3}$.
However (and here I'm probably mistaken), if I look at a side of the triangle, say $a$, by the triangle inequality it must be in the range $0<a<\frac{p}{2}$. Then, if the side length is uniformly distributed, this would imply that the average side length is $\frac{p}{4}$.
How can the second approach be corrected? (I assume the problem is that I wrongly assumed the distribution of $a$).
The reason I want to use the second approach is because more complex problems (for example, finding means of sizes that are a function of a side length) would require knowing how $a$ is distributed.