Volumes of Polyhedra

The exposition of the material begins with a formula expressing the volume of a tetrahedron in terms of the lengths of its edges. This formula can be found in almost all mathematical handbooks, but few know its history. The brochure discusses the proofs of this formula attributed to Tartaglia (16th century) and Euler (18th century), and provides modern variations of them. A theorem is formulated and commented upon, which generalizes the volume formula for tetrahedra to all polyhedra and gives as a simple corollary the solution to the problem of "blacksmith's bellows," asserting the constancy of the volume of a bent polyhedron. Examples of bendable polyhedra are also provided.
The text of the brochure represents an enhanced processing of a lecture note for 9th-11th grade students, delivered by the author at the Small Mechanics and Mathematics School of the Moscow State University on March 10, 2001 (recorded by E. A. Chernysheva).
The brochure is aimed at a broad audience interested in mathematics: high school seniors, junior college students, and teachers.