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.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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TERRELLA
TERRESTRIAL
TERTIATE
TETRACHORD
TETRADIAPASON
* TETRAEDRON
TETRAGON
TETRAGONIAS
TETRAGONISM
TETRASPASTON
TETRASTYLE