$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\om}{\omega} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathcal{F}} \newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\Var}{\operatorname{\mathsf Var}} \renewcommand{\P}{\operatorname{\mathsf P}} \newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}} \newcommand{\eD}{\overset{\text{D}}\to} \newcommand{\D}{\overset{\text{D}}=} \newcommand{\tsi}{\tilde\si}$ Let $z_0,z_1,z_2$ be the random points on the circle. We may assume that $z_0=1$, $z_1=e^{i\thh_1}$, $z_2=e^{i\thh_2}$, where $\thh_1,\thh_2$ are iid uniformly distributed on $[0,2\pi]$, and so, by the formula $A_\bigtriangleup=\frac12\,ab\sin C$, the area of the random triangle is \begin{equation} A=2\sin U_1\,\sin U_2\,\sin(U_1+U_2), \end{equation} where the $U_i:=\thh_i/2$'s are iid uniformly distributed on $[0,\pi]$. So, in view of the Euler formula $\sin\thh=(e^{i\thh}-e^{-i\thh})/(2i)$, all the moments \begin{equation} \E A^n=\frac1{\pi^2}\iint_{[0,\pi]^2}(2\sin u_1\,\sin u_2\,\sin(u_1+u_2))^n\,du_1\,du_2\, \end{equation} (with natural $n$) are rational numbers, which are actually diadic, with denominators being natural powers of $2$. This is so because the integrand is a polynomial in $e^{iu_1},e^{iu_2}$ with diadic rational coefficients. In particular, for $n=2,4,6$ we have the respective values $3/8,45/126,105/256$ of $\E A^n$.