Eisenstein integers and equilateral ideal triangles

We discuss the relationship between Penner’s $\lambda$-length and the norms of Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that every prime $p$ of the form $3k+1$ is the norm of an Eisenstein integer that is can be written as $a^2-ab+b^2$ for some $a,b\in \mathbb{Z}$.