You don't need to know the area separately, since that is given by the classic formula $$ |T| = (\gamma_1+\gamma_2+\gamma_3) - \pi. $$ Second, if $\ell_i$ is the length of the side opposite $\gamma_i$, then the standard spherical trig formula called the polar law of cosines gives $$ \ell_i = \cos^{-1}\left(\frac{\cos\gamma_i + \cos\gamma_j\cos\gamma_k}{ \sin\gamma_j\sin\gamma_k}\right) $$ where $(i,j,k)$ is a permutation of $(1,2,3)$. Now $P = \ell_1+\ell_2+\ell_3$, is such a formula.

First, you don't need to know the area separately, since that is given by the classic formula $$ |T| = (\gamma_1+\gamma_2+\gamma_3) - \pi. $$ Second, if $\ell_i$ is the length of the side opposite $\gamma_i$, then the standard spherical trig formula called the polar law of cosines gives $$ \ell_i = \cos^{-1}\left(\frac{\cos\gamma_i + \cos\gamma_j\cos\gamma_k}{ \sin\gamma_j\sin\gamma_k}\right) $$ where $(i,j,k)$ is a permutation of $(1,2,3)$. Now, $$P = \ell_1+\ell_2+\ell_3$$ is such a formula.