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{\bf Question}

Determine whether or not there exists a number $\alpha>0$, so that
there exists a hyperbolic triangle T whose interior angles are
$\frac{\pi}{3},\ \frac{\pi}{5}$, and $\alpha$, and whose
hyperbolic area area(T) is $\frac{\pi}{25}$. If such an $\alpha$
exists, determine its value (or values).
\medskip

{\bf Answer}

By the Gauss-Bonnet Formula:

${\rm{area}}(\tau)=\pi-$\ (sum of interior angles)

and so \begin{eqnarray*} {\rm{area}}(\tau) & = &
\pi-\left(\ds\frac{\pi}{3}+\ds\frac{\pi}{5}+\alpha\right)\\ & = &
\pi\left(1-\ds\frac{1}{3}-\ds\frac{1}{5}\right)-\alpha\\ & = &
\pi\ds\frac{7}{15}-\alpha \end{eqnarray*}

Since the only requirement is that area$(\tau)>0$, there is such
an $\alpha$, namely
$\alpha=\left(\ds\frac{7}{15}-\ds\frac{1}{25}\right)\pi =
\un{\ds\frac{32}{75}\pi}$

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