Probability of the surface area of a pentagonal prism being greater than the volume

Question

Consider a unit cube. Points $WXYZ$ are on the sides of the bottom face of the unit cube. Point $I$ is inside of the bottom face of the unit cube such that $WXYZI$ forms a pentagon. Point $M$ is somewhere inside of the unit cube. What is the probability that the surface area of the pentagonal pyramid with base $WXYZI$ and apex $M$ is greater than its volume?

I tried putting the problem in the coordinate plane and that helped me find the volume in terms of several variables, but I'm still not sure how to find the surface area of the pyramid. In addition, there are several different cases depending on the position of $I$, and I don't know how to find the probability of each case occurring. Thanks!

Answer 1

The surface area always exceeds the volume.

Proof.

Volume = $\frac{1}{3}$ (Area of Base) x (Perpendicular height)

$< \frac{1}{3}$ (Total Area Prism) since height of M is at most 1.

Answer 2

Does it specify whether it is a right or an oblique pyramid? For an oblique pyramid there is no formula for the lateral area since each lateral face can be different; they must be calculated separately and then added together.