Proposition XII.11 — Cones and cylinders which are of the same height are to one …

Proof

By XII.2 the bases (circles) are in the squared-diameter ratio; by the formula in XII.10, the volumes follow the same ratio.

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Depends on (2)

XII.2Proposition XII.2Circles are to one another as the squares on the diameters.
XII.10Proposition XII.10Any cone is a third part of the cylinder which has the same base with it and equal height.

Required by (dependents) (4)

XII.12Proposition XII.12Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
XII.13Proposition XII.13If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so…
XII.14Proposition XII.14Cones and cylinders which are on equal bases are to one another as their heights.
XII.15Proposition XII.15In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in…

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