Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume

(Proposition 31 from Book 11 of Euclid's “Elements”)

Parallelepiped solids which are on equal bases, and (have) the same height, are equal to one another.

Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2 3 4

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016