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:

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