Any convex 3D object will have a silhouette of a certain area. Averaged over all viewing angles, the size of the silhouette is exactly a quarter of the object's surface area.

Why? By dimensional analysis, the size of the silhouette must be proportional to surface area. By subdividing the surface into shadow-casting pieces, you can prove that the constant doesn't depend on shape as long as the shape is convex. So use the example of a unit sphere to compute its value.

The same principle applies in higher dimensions n. The formulas for sphere surface area and volume involve the gamma function, which simplifies if you consider odd and even dimensions separately. Let \(\lambda_n\) designate the constant for n dimensions. Then:

$$\lambda_{n+1} = \begin{cases}\frac{1}{2^{n+1}} {n \choose n/2} & n\text{ even}\\\frac{1}{\pi}\frac{2^n}{n+1} {n \choose {\lfloor n/2\rfloor} }^{-1} & n\text{ odd}\end{cases}$$