  Let $M$ be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane
$\HH^2 \subset \RR^3$. If $\Delta(abc)$ is a geodesic triangle on $M$ with
corners at $a,b,c\in M$, we denote by $\alpha, \beta, \gamma\in M$ the
midpoints of their sides. If $\Omega$ denotes the oriented area of this
triangle on $M$, it satisfies the relations:
  $$ \sin(\Omega/2) = \frac{\Det_3(abc)}{\sqrt{2(1 + \scalar{a}b)(1 +
\scalar{b}c)(1 + \scalar{c}a)}} = \Det_3(\alpha\beta\gamma) \, $$ where
$\scalar{\}{\}$ denotes the Euclidean scalar product for $M=\SS^2$ and the
Lorentzian scalar product for $M=\HH^2$. On the hyperbolic plane one should
always take the solution with $\modu{\Omega/2}<\pi/2$. On the sphere, singular
cases excepted, a straightforward procedure tells us which solution of this
equation is the correct one.
