The mean caliper formula was derived in 1967 by the metallurgist John Cahn [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 $\langle H\rangle$ is related to the average caliper $\langle C\rangle$ by $$\langle H\rangle=\frac{2\pi}{A}\left(\langle C\rangle-\frac{\epsilon}{4\pi}\sum_e L_e\right)$$ where $A$ is the total area of the faces and $L_e$ is the length of edge number $e$.
[1] J.W. Cahn, The significance of average mean curvature and its determination by quantitative metallography, Trans. Met. Soc. AIME 239 (1967) 610-616. behind a paywall