Mean projected area of a randomly oriented thin disc ($⟨A_{\mathrm{proj}}⟩=\pi R^2/2$)

A thin circular disc presents projected area $\pi R^2|\cos i|$ to a sightline. Isotropic normals have a uniform distribution of $\cos i$ on $[-1,1]$, giving mean absolute cosine one half. The resulting interception cross-section is therefore half the face-on area; no extra orientation weighting is needed after this average is used in an incidence calculation.