How to recalculate 2D polygon side lengths when tilt is applied in 3D space?

The height of a point after applying this transformation is a function of the distance to the axis fixed by this transformation. Specifically, draw the line $\ell$ through $C$ which remains unchanged after "tilting." Then if a point $P$ is a distance $d$ from $\ell$(to calculate the distance, drop a perpendicular line segment to $\ell$ at a point $Q$ and measure the length $PQ$), the height of $P$ after tilting is found through some trigonometry. Specifically, let $P'$ be the point after tilting. Then $QPP'$ is a right triangle with a $30^\circ$ angle. Thus $PP' = \tan(30^\circ)QP = d/\sqrt{3}$.

Consider points $A, B$ on the plane with distances $d_A, d_B$ to $\ell$. Let $A', B'$ be the points in $3$-space after tilting. Then
$$A'B' = \sqrt{AB^2 + (d_A - d_B)^2/3}.$$

This follows from the Pythagorean theorem, since translating the plane up to $A'$ creates a right triangle with legs of length $AB$ and $|d_A - d_B|$, with hypotenuse $A'B'$.

Thus to calculate the change in side length, one must additionally calculate the distances to the axis $\ell$ and use the above formula.

The relevant field of math used here is trigonometry and Euclidean geometry.