Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases

Euclid's Formulation

Cones and cylinders having the same height are to one another as their bases. * Let there be cones and cylinders of the same height whose bases [are] the circles $ABCD$ and $EFGH$, axes $KL$ and $MN$, and diameters of the bases $AC$ and $EG$ (respectively). * I say that as circle $ABCD$ is to circle $EFGH$, so cone $AL$ (is) to cone $EN$.


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2 3

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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