Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height

Euclid's Formulation

Every cone is the third part of the cylinder which has the same base as it, and an equal height. * For let there be a cone (with) the same base as a cylinder, (namely) the circle $ABCD$, and an equal height. * I say that the cone is the third part of the cylinder - that is to say, that the cylinder is three times the cone.


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