Fix any point $R$. Let $\phi_1$, $\phi_2$ and $\phi_3$ be angles, relative to $R$, then $0 < \phi_1 < \phi_2 < \phi_3 < 2 \pi$, i.e the triple $\{\phi_1, \phi_2, \phi_3\}$ is the first, second and the third order statistics of the uniformly distributed angle on the interval $(0, 2\pi)$.
The area of the triangle, is then $ A = 4 \sin \left( \frac{\phi_2-\phi_1}{2} \right) \sin \left( \frac{\phi_3-\phi_2}{2} \right) \sin \left( \frac{\phi_3-\phi_1}{2} \right) $ and the perimeter $ p = 2 \left( \sin \left( \frac{\phi_2-\phi_1}{2} \right) + \sin \left( \frac{\phi_3-\phi_2}{2} \right) + \sin \left( \frac{\phi_3-\phi_1}{2} \right) \right) $ Given that $\frac{\phi_3 -\phi_1}{2} = \frac{\phi_3 -\phi_2}{2} + \frac{\phi_2 -\phi_1}{2}$, let $\alpha = \frac{\phi_2-\phi_1}{2}$ and $\beta = \frac{\phi_3-\phi_2}{2}$, then $ A(\alpha, \beta) = 4 \sin \alpha \sin \beta \sin (\alpha+\beta) \qquad p(\alpha, \beta) = 2 \left( \sin \alpha + \sin \beta + \sin (\alpha+\beta) \right) $ The distribution of $\alpha$ and $\beta$ are not hard to find. Indeed, distribution of $\phi_1$, $\phi_2$ and $\phi_3$ ($[\chi]$ stands for Iverson bracket): $ \mathrm{d} F(\phi_1, \phi_2, \phi_3) = \frac{3!}{(2 \pi)^3} \Big[ 0 < \phi_1 < \phi_2 < \phi_3 < 2 \pi \Big] \mathrm{d} \phi_1 \mathrm{d} \phi_2 \mathrm{d} \phi_3 $ Changing variables to $\alpha = \frac{\phi_2-\phi_1}{2}$, $\beta=\frac{\phi_3-\phi_2}{2}$ and $\gamma = \frac{1}{3} \left( \phi_1 + \phi_2 + \phi_3 \right)$ we get, noting that the Jacobian equals $4$, $ \mathrm{d} F(\alpha, \beta, \gamma) = \frac{3!}{(2 \pi)^3} \Big[ 0 < \alpha < \pi, 0 < \beta < \pi-\alpha, \frac{4 \alpha + 2 \beta}{3} <\gamma< 2\pi - \frac{2 \alpha + 4 \beta}{3} \Big] \cdot 4 \cdot \mathrm{d} \alpha \, \mathrm{d} \beta \, \mathrm{d} \gamma $ Integrating over $\gamma$ we find the joint pdf for $(\alpha, \beta)$: $ \mathrm{d} F\left(\alpha,\beta\right) = \frac{3!}{\pi^3} \left( \pi - \alpha - \beta \right) \Big[ 0 < \alpha < \pi, 0< \beta < \pi - \alpha \Big] \, \, \mathrm{d} \alpha \, \mathrm{d} \beta $ This means that $\left\{ \frac{\alpha}{\pi}, \frac{\beta}{\pi} \right\}$ follows Dirichlet distribution with parameters $\{1,1,2\}$.

Using the interpretation in terms of the Dirichlet distribution, it is not hard to determine the mean and the variance of the area and the perimeter:
