Cube to a Low Pyramid, and Pyramid to a Cube
Problems 7 and 8: comparing the bases through slab and cylinder
This leaf carries a run of paired stereometry problems. From a cube, make a pyramid at a given height lower than three heights of the cube, asking how much wider the pyramid's base grows; the solution divides the pyramid's height into three, reduces the cube to one such part, and turns it through cylinder into cube to compare the bases. The final problem reverses direction: from a pyramid lower than three sides of its base, make a cube, by first forming a slab a third as tall as the pyramid and as wide as its base, then a cylinder, then the requested cube. Three small solids, an outline cube, a pyramid-in-cube, and a pyramid on a slab, are sketched down the right margin.
On this page
From a cube to a low pyramid
A cube is to become a pyramid at a given height lower than three heights of the cube, and the question is how much wider the pyramid's base is than the cube's base. An outline cube is drawn at the upper right to set the problem.
Solving through slab and cylinder
The height given to the pyramid is divided into 3 parts and the cube reduced to the height of one such part; by the fifth of the first it becomes its natural cylinder, and by the first of the second a cube. The amount the slab's base exceeds the cube then equals the amount the pyramid's base exceeds the cube's base.
A low pyramid back to a cube
For a pyramid lower than three sides of its base joined on one line, a cube is made: a slab a third as tall as the pyramid and as wide as its base is formed, turned to a cylinder by the sixth of the first, and to the requested cube by the first of the second, asking how much narrower that cube's base is.