27. Projection and Cross-Section

Let $C$ be a cube and $S$ be a sphere in $\mathbb{R}^3$.

For each of the two solids, investigate and compare the following operations with an arbitrary plane $\Pi$:

1. The orthogonal projection of the solid onto $\Pi$.
2. The cross-section obtained by intersecting the solid with $\Pi$.

Determine the possible shapes and areas that can arise in each case, and establish how the projection and cross-section differ geometrically.

In particular, determine the maximum possible area of a projection and of a cross-section for both the cube and the sphere.

Justify all of your results.