Cauchy’s polyhedron formula

Cauchy’s polyhedron formula states that for a convex polyhedron, the surface area can be calculated by averaging the areas of all possible "views" of its faces from every direction around the shape. Imagine shining light in all directions and observing the silhouette; the total surface area relates to these observed cross-sections. Mathematically, it involves integrating the areas of these projections over the entire sphere of directions, connecting the shape’s geometry with its directional shadows. This theorem provides a powerful way to determine surface area without measuring each face directly, especially useful for complex convex shapes.