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In analyzing polyhedra I will do so without the use of trigonometry, unless calculating central or surface angles.

I will use the pyramid method for calculating volumes.

The volume of any n-sided pyramid = 1/3 * area of base * pyramid height.

All of the 5 regular solids have equal sides and equal angles.

Tetrahedron

Octahedron

Cube

Icosahedron

Dodecahedron

Now that we have learned something about these solids, lets
summarize
our knowledge:

Centroid to: | Centriod to: | Centroid to: | |||||||||

V(s = 1) s³ | V(r = 1) r³ | Surface Area (s=1) s² | Surface Area (r=1) r ² | Central < | Dihedral < | Surface < | Vertex | Mid-edge | Mid-face | side /radius | |

Tetrahedron | 0.11785113 | 0.513200239 | 1.73205 | 4.618802154 | 109.4712206° | 70.52877936° | 60° | 0.612372436 | 0.3535534 | 0.204124 | 1.632993162 |

Octahedron | 0.47140452 | 1.333.... | 3.46410 | 6.92820323 | 90° | 109.4712206° | 60° | 0.707106781 | 0.5 | 0.408248 | 1.414213562 |

Cube | 1.0 | 1.539600716 | 6.0 | 8.0 | 70.52877936° | 90° | 90° | 0.866025404 | 0.707106781 | 0.5 | 1.154700538 |

Icosahedron | 2.181694991 | 2.53615071 | 8.660254 | 9.57454138 | 63.4349488° | 138.1896852° | 60° | 0.951056517 | 0.809016995 | 0.755761314 | 1.05146222 |

Dodecahedron | 7.663118963 | 2.785163863 | 20.64573 | 10.51462224 | 41.81031488° | 116.5650512° | 108° | 1.401258539 | 1.309016995 | 1.113616365 | 0.713644179 |

Here is the same table without all of the repeating decimals:

Note: Ø = 1.618033989, or (\/¯5 +
1) / 2, the Golden Section.

Centroid to: | Centroid to: | Centroid to: | |||||||||

V(s = 1) s³ | V(r = 1) r³ | Surface Area (s=1) s² | Surface Area (r=1) r ² | Central < | Dihedral < | Surface < | Vertex | Mid - Edge | Mid - Face | side / radius | |

Tetrahedron | 1 / 6\/¯2 | 8 / 9\/¯3 | \/¯3 | 8 / \/¯3 | 109.4712206° | 70.52877936° | 60° | \/¯3 / (2\/¯2) | 1 / (2\/¯2) | 1 / (2\/¯6) | 2\/¯2/ \/¯3 |

Octahedron | 2 / 3\/¯2 | 4 / 3 | 2 \/¯3 | 4 \/¯3 | 90° | 109.4712206° | 60° | 1 / \/¯2 | 1/2 | 1 / \/¯6 | \/¯2 |

Cube | 1.0 | 8 / 3\/¯3 | 6.0 | 8.0 | 70.52877936° | 90° | 90° | \/¯3 / 2 | 1 / \/¯2 | 1 / 2 | 2 / \/¯3 |

Icosahedron | 5Ø² / 6 | 20Ø² / 3(Ø²+1)^3/2 | 5 \/¯3 | 20\/¯3 /
(Ø²+1) |
63.4349488° | 138.1896852° | 60° | \/¯(Ø²+1)/ 2 | Ø / 2 | Ø² / (2 \/¯3) | 2 / \/¯(Ø²+1) |

Dodecahedron | 5Ø^5 /
2(Ø²+1) |
20Ø² / (3\/¯3)(Ø²+1) | 15Ø² / \/¯(Ø²+1) | 20 /
\/¯(Ø²+1) |
41.81031488° | 116.5650512° | 108° | (\/¯3)Ø /
2 |
Ø² / 2 | Ø³ /
2\/¯(Ø²+1) |
2 / (\/¯3)Ø |

From this chart we can see a number of things:

1) I have included 2 different ways to measure these
solids,
both of them helpful: the first keeping length of each side equal, and
the second considering each solid inscribed in a sphere of equal
radius.
The side/radius column describes the relationship between the
Volumes
and the Surface Areas for each solid. For Volume, use (s/r)³
when converting between V(s) and V(r).

For Surface Area, use (s/r)². For s = k or r = k, the Volume
will always increase k times as fast as the Surface Area.

2) The tetrahedron has the least volume with the most surface area,
the dodecahedron has the most volume with the least surface area. The
ratio
of the volume to surface area goes up as we go down the chart.

3) The central angles decrease as we go down the chart. This makes
sense as there are more vertices, so less space between the vertices.

4) The ratio of side to radius also decreases as the volume increases.
This also makes sense as in order to get more volume, the radius has to
increase. As the radius increases, in the solids with more vertices
there
is less surface area on the sphere over which to spread out.

5) the central < of the tetra = dihedral < of octa, the central
< of the octa = the dihedral < of the cube. So the octa is
created
from the tetra by using the same equilateral triangles as faces, and
applying
the tetra central angle to it. And so for the cube. This is consonant
with
the geometry of these solids, which is \/¯2 and
\/¯3
geometry.

This pattern breaks down when we get to the icosahedron and the
dodecahedron,
which are \/¯5 geometry.

Special characters:

\/¯ ° ¹ ² ³ × ½
¼
Ø \/¯(Ø² + 1)