Question #40778

Let $\hat{u}, \hat{v}, \hat{w}$ be three mutually orthogonal unit vectors in $\mathbb{R}^3$. Let $S_1, S_2,$ and $S_3$ be sets of points defined as follows: $S_1 = \{ P \in \mathbb{R}^3 : |\vec{OP}| = 2 \text{ and } |\vec{OP} \cdot \hat{u}| = 2 \}$ $S_2 = \{ Q \in \mathbb{R}^3 : |\vec{OQ}| = 3 \text{ and } |\vec{OQ} \cdot \hat{v}| = 3 \}$ $S_3 = \{ R \in \mathbb{R}^3 : |\vec{OR}| = 6 \text{ and } |\vec{OR} \cdot \hat{w}| = 6 \}$ where $O$ is the origin. Let $K$ be the convex polyhedron whose vertices are all the points in $S_1 \cup S_2 \cup S_3$. Find the sum of the squares of the areas of all the faces of $K$.