The Formula for the Volume of a Truncated Pyramid

Abstract

One of the current tasks of the modern Ukrainian school is the formation of students' mathematical competence, mainly geometric competence. Various approaches to proving theorems and formulas of geometry contribute to the development of geometric competence of high school students.

This article presents one of the proofs of the formula for the volume of a truncated pyramid. The formula for the volume of a truncated pyramid is an important result in solid geometry and its applications. In modern geometry textbooks, this formula is obtained as the difference of the volumes of two similar pyramids. The proof of this formula proposed in this article is based on the lemma about the division of a truncated tetrahedron into three tetrahedrons in such a way that the volume of one of such tetrahedrons is equal to the average proportional volume of the other two tetrahedrons. The proof of this lemma is also given in this article. Using this lemma, we obtain a formula for calculating the volume of a truncated tetrahedron. The transition to a truncated pyramid is accomplished by breaking the truncated pyramid into truncated tetrahedron.