A purely 3-D geometrical solution to Mathematics Magazine Problem 2065

We proposed a purely 3-D geometrical solution to Mathematics Magazine Problem 2065. Let $\mathcal{Q}$ be a cube centered at the origin of $\mathbb{R}^3$. Choose a unit vector $(a,b,c)$ uniformly at random on the surface of the unit sphere $a^2+b^2+c^2=1$, and let $\Pi$ be the plane $ax+by+cz=0$ through the origin and normal to $(a,b,c)$. What is the probability that the intersection of $\Pi$ with $\mathcal{Q}$ is a hexagon?