<P> In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by 1 3 ∑ F (Q F ⋅ N F) area ⁡ (F), (\ displaystyle (\ frac (1) (3)) \ left \ sum _ (F) (Q_ (F) \ cdot N_ (F)) \ operatorname (area) (F) \ right,) where the sum is over faces F of the polyhedron, Q is an arbitrary point on face F, N is the unit vector perpendicular to F pointing outside the solid, and the multiplication dot is the dot product . Since it may be difficult to enumerate the faces, volume computation may be challenging, and hence there exist specialized algorithms to determine the volume (many of these generalize to convex polytopes in higher dimensions). </P> <P> In two dimensions, the Bolyai--Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them . The analogous question for polyhedra was the subject of Hilbert's third problem . Max Dehn solved this problem by showing that, unlike in the 2 - D case, there exist polyhedra of the same volume that cannot be cut into smaller polyhedra and reassembled into each other . To prove this Dehn discovered another value associated with a polyhedron, the Dehn invariant, such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant . It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other . The Dehn invariant is not a number, but a vector in an infinite - dimensional vector space . </P> <P> Another of Hilbert's problems, Hilbert's 18th problem, concerns (among other things) polyhedra that tile space . Every such polyhedron must have Dehn invariant zero . The Dehn invariant has also conjecturally been connected to flexible polyhedra by the strong bellows conjecture, which asserts that the Dehn invariant of any flexible polyhedron must remain invariant as it flexes . </P> <P> A three - dimensional solid is a convex set if it contains every line segment connecting two of its points . A convex polyhedron is a polyhedron that, as a solid, forms a convex set . A convex polyhedron can also be defined as a bounded intersection of finitely many half - spaces, or as the convex hull of finitely many points . </P>

Is it possible to classify a polyhedron given only the number of faces