DODECAHEDRON

, one of the Platonic bodies, or five regular solids, being contained under a surface composed of twelve equal and regular pentagons.

To form a Dodecahedron. See Regular Body.

If the side, or linear edge, of a Dodecahedron be s, its surface will be and its solidity

If the radius of the sphere that circumscribes a Dodecahedron be r, then is its side or linear edge , its superficies , and its solidity .

The side of a Dodecahedron inscribed in a sphere, is | equal to the greater part of the side of a cube inscribed in the same sphere, and cut according to extreme and mean proportion.

If a line be cut according to extreme and mean proportion, and the lesser segment be taken for the side of a Dodecahedron, the greater segment will be the side of a cube inscribed in the same sphere.

The side of the cube is equal to the right line which subtends the angle of a pentagon of the Dodecahedron, inscribed in the same sphere. See Polyhedron; also my Mensur. pa. 253, &c.