Let $k \in \mathbb{R}$, and let $M_k^2$ denote the complete simply connected two-dimensional Riemannian model space of constant sectional curvature $k$. Let $\bar{p}, \bar{q}, \bar{r} \in M_k^2$ form a geodesic triangle with side lengths

\begin{align*} a &= d(\bar{q}, \bar{r}),& b &= d(\bar{p}, \bar{r}),& c &= d(\bar{p}, \bar{q}), \end{align*}

and let $\alpha \in [0,\pi]$ be the angle at $\bar{p}$ between the geodesic segments from $\bar{p}$ to $\bar{q}$ and from $\bar{p}$ to $\bar{r}$. Then:

\begin{align*} a^2 &= b^2 + c^2 - 2bc\cos \alpha &&\text{if } k = 0,\\ \cos(\sqrt{k}\,a) &= \cos(\sqrt{k}\,b)\cos(\sqrt{k}\,c) + \sin(\sqrt{k}\,b)\sin(\sqrt{k}\,c)\cos \alpha &&\text{if } k > 0,\\ \cosh(\sqrt{-k}\,a) &= \cosh(\sqrt{-k}\,b)\cosh(\sqrt{-k}\,c) - \sinh(\sqrt{-k}\,b)\sinh(\sqrt{-k}\,c)\cos \alpha &&\text{if } k < 0. \end{align*}