Solid geometry

geometry of three-dimensional Euclidean space

Solid geometry is the geometry of three-dimensional Euclidean space. It includes the measurements of volumes of various solid figures (three-dimensional figures). These include pyramids, cylinders, cones, spheres, and prisms.

It is Euclidean geometry, but not plane geometry. Points are defined as a defined position on a sphere. A "staight line" is the shortest path between two points which stays on the surface of the sphere.

Euclid's account of spherical geometry is in his Elements volumes XI–XIII containing solid geometry, and in his lesser-known work the Phaenomena, which includes 25 geometric propositions.[1][2] The actual discoveries were often made by others.

The Pythagoreans dealt with the regular solids, like the cube and the sphere. The pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus proved the pyramid and cone have one-third the volume of a prism and cylinder on the same base and of the same height.

The equations for areas and volumes in two and three dimensions was proved by Archimedes. One of Archimedes' works was called On the sphere and the cylinder. He asked that a carving of the two solid figures be placed on his tomb. He specified a "right circular cylinder with height equal to its circumference" because he had used this in his proof.[3]

People use solid geometry for many purposes including engineering and the calculation of perspective.


  1. Smith D.E. [1923] 1951. History of mathematics, volume 1. New York: Dover, p106.
  2. Boyer, Carl B. 1985. A history of mathematics. Princeton University Press, p129 section 12.
  3. Explained further in Boyer p 145.