It is not difficult for a class to find the relative areas of the cube and the inscribed tetrahedron and octahedron. If s is the side of the cube, these areas are 6s2, (1/2)s2√3, and s2√3; that is, the area of the octahedron is twice that of the tetrahedron inscribed in the cube.
Somewhat related to the preceding paragraph is the fact that the edges of the five regular solids are incommensurable with the radius of the circumscribed sphere. This fact seems to have been known to the Greeks, perhaps
to Theætetus (ca. 400 B.C.) and Aristæus (ca. 300 B.C.), both of whom wrote on incommensurables.
Just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon, so we may have stellar polyhedrons. Kepler, the great astronomer, constructed some of these solids in 1619, and Poinsot, a French mathematician, carried the constructions so far in 1801 that several of these stellar polyhedrons are known as Poinsot solids. There is a very extensive literature upon this subject.
The following table may be of some service in assigning problems in mensuration in connection with the regular polyhedrons, although some of the formulas are too difficult for beginners to prove. In the table e = edge of the polyhedron, r = radius of circumscribed sphere, r' = radius of inscribed sphere, a = total area, v = volume.
| Number of Faces | 4 | 6 | 8 | 12 | 20 |
| r | e√(3/8) | (e/2)√3 | e√(1/2) | (e/4)√3(√5 + 1) | e√((5 + √5)/8) |
| r' | e√(1/24) | e/2 | e√(1/6) | (e/2)√((25 + 11√5)/10) | (e√3)/12(√5 + 3) |
| a | e2√3 | 6e2 | 2e2√3 | 3e2√(5(5 + 2√5)) | (5e2)√3 |
| v | (e3/12)√2 | e3 | (e3/3)√2 | ((e3)/4)(15 + 7√5) | ((5e3)/12)(√5 + 3) |
Some interest is added to the study of polyhedrons by calling attention to their occurrence in nature, in the form of crystals. The computation of the surfaces and volumes of these forms offers an opportunity for applying the rules of mensuration, and the construction of the solids by paper folding or by the cutting of crayon or some other substance often arouses a considerable interest. The following are forms of crystals that are occasionally found:
They show how the cube is modified by having its corners cut off. A cube may be inscribed in an octahedron, its vertices being at the centers of the faces of the octahedron. If we think of the cube as expanding, the faces of the octahedron will cut off the corners of the cube as seen in the first figure, leaving the cube as shown in the second figure. If the corners are cut off still more, we have the third figure.