Geometry: Pyramids, the Cube's Three Roots, and Pythagoras
Forming a pyramid on a given base, the three square roots within a cube, and a Pythagorean figure
A geometry page posing solid problems with irrational quantities: from an extensible quantity, form a pyramid on a given base and find its height, or form one of a given height and find its thickness. A cube diagram illustrates 'the three roots in the cube' - the roots of squares equal to a sixth, a third, and a half of the cube's surface. A separate figure sets out the Pythagorean theorem with squares numbered on the sides of a right triangle.
On this page
Forming a pyramid from an extensible quantity
From a given extensible quantity, make a pyramid of thickness set by a given base and find its height; or make one whose height equals a given line and find its thickness. All the given and received quantities are irrational.
The three roots contained in the cube
The first root a b is the root of the square a b e f, a sixth of the cube's surface. The second root a c is the root of a square double that, equal to a third of the surface; the third root a d is the root of a square triple the first, equal to half the surface.
Figure of the Pythagorean theorem
A right triangle carries squares on its sides, labelled 3, 'first', and 2, to display the Pythagorean relation among the areas.