The mean caliper formula was derived in 1967 by the metallurgist <A HREF="https://en.wikipedia.org/wiki/John_W._Cahn">John Cahn</A> [1]. Cahn allowed for a curvature $K$ of the faces and assumed that all dihedral angles $\pi-\delta_e$ are the same, equal to $\epsilon$. Then the average curvature $\langle K\rangle$ is related to the average caliper $\langle C\rangle$ by
$$\langle K\rangle=\frac{2\pi}{A}\left(\langle C\rangle-\frac{\epsilon}{4\pi}\sum_e L_e\right)$$
where $A$ is the total area of the faces and $L_e$ is the length of edge number $e$.

I don't have access to the paper, but presumably with this info it can either be located in some library, or the content is reproduced elsewhere.

[1] J.W. Cahn, *The significance of average mean curvature and its determination by quantitative metallography*, Trans. Met. Soc. AIME 239 (1967) 610-616.
<sub>behind a <A HREF="http://www.aimehq.org/resources/library">paywall</A></sub>