Coordinate variance in a 10-dimensional ball

A point $(X_1,\dots,X_{10})$ is chosen uniformly from the solid ball of radius $12$ centred at the origin in $\mathbb{R}^{10}$. What is $\text{Var}(X_1)$?

$12$. By symmetry $\mathbb{E}[X_1]=0$ and $\mathbb{E}[X_1^2]=\tfrac1{10}\mathbb{E}[R^2]$. The radius has $\mathbb{P}(R\le r)=(r/12)^{10}$, so $\mathbb{E}[R^2]=\tfrac{10}{12}\cdot144=120$, giving variance $12$.