Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? $\mathrm {(A)}\ \sqrt{2} \qquad$ $ \mathrm {(B)}\ \sqrt{3} \qquad$ $ \mathrm {(C)}\ 1+\sqrt{2}\qquad$ $ \mathrm {(D)}\ 1+\sqrt{3}\qquad$ $ \mathrm {(E)}\ 3$
Solution
Each small sphere (radius $1$) tangent to all three coordinate planes has its center at $(\pm1,\pm1,\pm1)$, at distance $\sqrt3$ from the origin.
The smallest containing sphere centered at the origin must reach the farthest point of each small sphere, at distance $\sqrt3+1$.
Four spheres of radius $1$ are placed so that each is tangent to the other three. What is the radius of the smallest sphere that contains all four of them?
$\mathrm {(A)}\ \sqrt{2} \qquad$ $ \mathrm {(B)}\ \sqrt{3} \qquad$ $ \mathrm {(C)}\ 1+\sqrt{2}\qquad$ $ \mathrm {(D)}\ 1+\sqrt{3}\qquad$ $ \mathrm {(E)}\ 3$ 