I have previously asked a similar question, but that turned out to be complicated. Here is a related simpler question.

Let $K$ be a convex set around the origin in $\mathbb R^3$. $Z(\theta)$ is a point on $K$ in the direction $\hat \theta$. $h(\theta)$ is the perpendicular distance from origin to the tangent plane of $K$ at $Z(\theta)$. Can we write down the volume of the convex set in terms of $h(\theta)$. Actually the volume($V$) is : $V=\frac{1}{3} \int ds \\ h(\theta)$, where $ds$ is the area element $K$ in the $\hat \theta$ direction. Question is what is the relation between $ds$ and $d\theta$ in terms of $h(\theta)$.

In two dimensions: $ds=(h+\ddot{h}) d\theta$