Let \(\Gamma= \text{PSL}_2(\mathbb{Z})\) and \(\Gamma_\infty=\{\gamma\in\Gamma\mid \gamma\infty=\infty\}\). The authors prove a tessellation of the half-plane \(H:= \{(x,y)\in\mathbb{R}^2\mid y\geq 1\}\) into semi-infinite triangles with disjoint interiors \(H= \bigcup_{\gamma\in\Gamma\setminus\Gamma_\infty} \Delta(\gamma)\), where for \(\gamma= \begin{pmatrix} a & b\\ c & d\end{pmatrix}\) with \(c>0\) \[ \Delta(\gamma)= \{(x,y)\in\mathbb{R}^2\mid 0\leq d-cx-ay\leq c\leq -dx-by\}. \] An application of this tessellation is a proof of a refinement of the Kronecker-Hurwitz class number relation.