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Volume of a rectangular parallelepiped

Lecture



In order to find the volume of a rectangular parallelepiped with linear dimensions a, b, c, we prove that the volumes of two rectangular parallelepipeds with equal bases are referred to as their heights.

  Volume of a rectangular parallelepiped

Let P and P1 be two rectangular parallelepipeds with a common base ABCD and heights AE and AE1. We assume for definiteness that AE1 <AE. Let V and V1 be the volumes of parallelepipeds. Divide the edge AE of the parallelepiped P into a large number n equal parts. Each of them is equal to AE / n. Let m be the number of division points that lie on the edge AE1. Then

  Volume of a rectangular parallelepiped

From here

  Volume of a rectangular parallelepiped

Draw through the dividing points of the plane parallel to the base. They will divide the parallelepiped P into n equal parallelepipeds. Each of them has a volume V / n. The parallelepiped P1 contains the first m parallelepipeds, counting from below, and is contained in m + 1 parallelepipeds. therefore

  Volume of a rectangular parallelepiped

From here

  Volume of a rectangular parallelepiped

Since V1 / V and AE1 / AE are between m / n and m / n + 1 / n, they differ by no more than 1 / n. And since n can be taken arbitrarily large, it can only be at

  Volume of a rectangular parallelepiped

Q.E.D.

We now take a cube, which is a unit of measurement for volume, and three rectangular parallelepipeds with dimensions: a, 1, 1; a, b, 1; a, b, c. We denote their volumes V1, V2 and V, respectively. According to proven

  Volume of a rectangular parallelepiped

Multiplying these equalities term by term, we get: V = abc
created: 2014-10-05
updated: 2021-03-13
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Stereometry

Terms: Stereometry