TETRAEDRON
, or Tetrahedron, in Geometry, is one of the five Platonic or regular bodies or solids, comprehended under four equilateral and equal triangles. Or it is a triangular pyramid of four equal and equilateral faces.
It is demonstrated in geometry, that the side of a Tetraedron is to the diameter of its circumscribing sphere, as √2 to √3; consequently they are incommensurable.
If a denote the linear edge or side of a Tetraedron, b its whole superficies, c its solidity, r the radius of its inscribed sphere, and R the radius of its circumscribing sphere; then the general relation among all these is expressed by the following equations, viz,
| a = 2r√6 | = (2/3)R√6 | = √((1/3)b√3) | = √3(6c√2). |
| b = 24r2√3 | = (8/3)R2√3 | = a2√3 | = 6√3(c2√3). |
| c = 8r3√3 | = (8/27)R3√3 | = (1/12)a3√2 | = (1/36)b√(2b√3). |
| R = 3r | = (1/4)a√6 | = (1/4)√(2b√3) | = (3/2)√3((1/3)c√3). |
| r = 1/3R | = (1/12)a√6 | = (1/12)√(2b√3) | = (1/2)√3((1/3)c√3). |
See my Mensuration, pa. 248 &c, 2d ed. See also the articles Regular and Bodies.
