A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

130 Polyhedra.
described, cutting 0A, OE, OC in the points a, e, c, respectively; then aec is a spherical triangle, having the angle e right;
77 7r
also cae = and ace = -, and by equation (111), sin ace = cos cae
n m
cos ce; but cos ce = cos coe = sin I;
7r 7r. sin I= cosec- cos -.
m n
Cor. 1.-The following are the values of Ifor the five regular
polyhedra. Thus, denoting them by P4, P6, P8, P12, P20:1 7r i
In P4, cos=-; in P6, I= in P8, cos I= - -
3 2; 3'
in P2, cos I-; in P20, cos = - /5.
-/5 3
Cor. 2.-If r be the radius of the inscribed sphere, and a
a side of one of the faces,
a 7T i
r= cot-. tan. (429)
2 in 2'
I a r I
For r = CE. tan CEO = CEtan cot - tan
2 2 m 2
Cor. 3.-If R be the circumradius of the polyhedron,
a ir I
R= - tan- tan-. (430)
2 n 2
Cor. 4.-The surface of a regular polyhedron
ma2 F 77
= maFcot -. (431)
4 m
Cor. 5.-The volume of a regular polyhedron
ma2 rF 7r
= 12 cot (432)
12 c
Cor. 6.-The octahedron is the reciprocal of the cube, and
the icosahedron of the dodecahedron.