Divisions of Solids: Pyramids Within Cubes
Proving the volumes of pyramids, cubes and their sections by proportion
Headed "Divisions of solids," this page pairs five lettered diagrams with proofs about the volumes of pyramids and cubes. Leonardo shows that a cube resolves into six equal pyramids meeting at its center, that the largest pyramid drawn from a cube equals a third of it, and that half a pyramid's height gives an eighth of its volume. Further propositions establish that the largest cube inscribed in a square pyramid is three eighths of it, and that a diagonally cut cube equals its inscribed pyramid.
On this page
A cube resolves into six equal pyramids
Leonardo divides the cube into six pyramids equal among themselves, all meeting at the central point o: a b c d o, a b h e o, h e g f o, g f c d o, h g c a o, and b e d f o. The construction underlies the volume proofs that follow.
The largest pyramid drawn from a cube is a third of it
The greatest pyramid that can be drawn from a cube is worth a third of that cube. In the second figure the larger pyramid a c h g d has no pyramid opposite it on the opposite base, so it alone holds the value of two of the six pyramids, that is a third of the cube.
Half a pyramid's height gives an eighth of its volume
Under the heading "third," Leonardo states that the half of the height of any pyramid is worth the eighth part of the whole pyramid, as proved in his book of pyramids. It is a proportional rule linking a linear halving to a cubic ratio.
The largest cube in a pyramid is three eighths of it
The greatest cube that can be drawn from a quadrilateral pyramid is worth three eighths of the whole pyramid. Leonardo derives this from the earlier proposition that the largest pyramid drawn from a cube is a third of that cube, reasoning through the pyramid a b c d e set on the enclosed cube.