Orthogonal projection area formula ($d{A}_{\mathrm{proj}}=|\mathbf{n}\cdot \mathbf{k}|dA$)

For projection of a planar region with unit normal $\mathbf{n}$ onto a plane with unit normal $\mathbf{k}$, the projected area is $|\mathbf{n}\cdot \mathbf{k}|$ times the original area. Indeed, the signed projected area of the parallelogram spanned by tangent vectors $\mathbf{u},\mathbf{v}$ is $(\mathbf{u}\times \mathbf{v})\cdot \mathbf{k}$, while $\mathbf{u}\times \mathbf{v}$ is parallel to $\mathbf{n}$. Taking magnitudes proves the factor. The same pointwise formula applies to a parametrized surface; adding projected patches counts multiplicity unless their images overlap only in sets of zero area.