Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases

Euclid's Formulation

Similar cones and cylinders are to one another in the cubed ratio of the diameters of their bases. * Let there be similar cones and cylinders of which the bases (are) the circles $ABCD$ and $EFGH$, the diameters of the bases (are) $BD$ and $FH$, and the axes of the cones and cylinders (are) $KL$ and $MN$ (respectively). * I say that the cone whose base [is] [circle]bookofproofs$690 $ABCD$, and apex the point $L$, has to the cone whose base [is] [circle]bookofproofs$690 $EFGH$, and apex the point $N$, the cubed ratio that $BD$ (has) to $FH$.


Modern Formulation

(not yet contributed)

Proofs: 1

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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