Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides

Euclid's Formulation

Similar pyramids which also have triangular bases are in the cubed ratio of their corresponding sides. * Let there be similar, and similarly laid out, pyramids whose bases are triangles $ABC$ and $DEF$, and apexes the points $G$ and $H$ (respectively). * I say that pyramid $ABCG$ has to pyramid $DEFH$ the cubed ratio of that $BC$ (has) to $EF$.


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