Converting a cylinder to a tall pyramid, and pyramid to cylinder
Problems 11 and 12 (c, d, e; a, b): equal-volume conversions between cylinder and pyramid
Two numbered problems (11 and 12) convert between cylinder and pyramid while keeping the same quantity of material. Problem 11 makes a pyramid from a cylinder so tall that its length exceeds three of the cylinder's heights joined together, worked in the same way as the earlier tenth problem. Problem 12 does the reverse for pyramid c d e, transmuting it into its natural cylinder and then, by the fifth of the second, stretching that into a longer cylinder to reach the required height a b. Diagrams of hanging pyramids and a pyramid inside a cylinder illustrate the two cases.
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Problem 11: a tall pyramid from a cylinder
From a cylinder a pyramid is to be made at a given height such that its length exceeds three heights of that cylinder joined together. Leonardo notes that this eleventh problem is carried out exactly as the tenth was done.
Problem 12: a cylinder from pyramid c d e to height a b
For pyramid c d e, the given cylinder height into which its quantity is transmuted is a b. It is first turned into its natural cylinder by the twelfth of this third book, then by the fifth of the second made into a longer cylinder extending to the required height, and so it is done.