How to Calculate $\operatorname{Var}(X)$?

Question

We choose a point $Y$ on pencil of length 1, S.T $Y\sim \operatorname{Uni}(0,1)$

We break the pencil at that point, choose one of the 2 parts in equal probability (1/2), S.T $X$ is the length of the part we chose. ie the length of the other part is $1-X$.

1. Calculate $\mathrm E(X)$.
2. Calculate $\operatorname{Var}(X)$.

For (1) I found that $X=Y$ or $X=1-Y$ in both cases where probability is 1/2 so the answer will be 1/2.

For (2) I know, $\operatorname{Var}(X)=\mathrm E(X^2)-\mathrm E(X)^2=\mathrm E(X^2)-1/4$ But How to Continue from here?

Answer 1

Since $X$ is uniform in the interval $[0,1]$, $E(X^2)=\int_0^1x^2dx=\frac{1}{3}$. Variance=$\frac{1}{12}$.