Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides

(Proposition 33 from Book 11 of Euclid's “Elements”)

Similar parallelepiped solids are to one another as the cubed ratio of their corresponding sides. * Let $AB$ and $CD$ be similar parallelepiped solids, and let $AE$ correspond to $CF$. * I say that solid $AB$ has to solid $CD$ the cubed ratio that $AE$ (has) to $CF$.


Modern Formulation

(not yet contributed)

Proofs: 1 Corollaries: 1

Proofs: 1 2

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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