Archive for January, 2015

Icosahedron

2015-01-29
Icosahedron

The icosahedron has 12 equilateral triangles as faces. It can be split into 3 parts, a pentagonal anti-prism and two pentagonal pyramids. We will start by looking at a pentagonal pyramid. We have already done most of the work earlier. To find ∠ACB, $$\begin{align}\cos \angle …Read the Rest

Dodecahedron

2015-01-28
Dodecahedron

The dodecahedron is a regular polyhedron made of 12 regular pentagon faces. A D12 die is often used in certain role playing games. Since a dodecahedron has 12 sides, you can print out a calendar, one month per face, and have a conversation piece on …Read the Rest

Pentagon

2015-01-02
Pentagon

A regular pentagon has the following dimensions: AB=BC=CD=DE=AE=BG=EG=1 Height AH=EK=\(\frac12\cos 18 = \frac12\sqrt{5+2\sqrt5}\) AC=AD=BD=BE=CE=Φ Circumcenter AJ=\(\frac{\sqrt{50+10\sqrt5}}{10}\)=BJ=CJ=DJ=EJ BF=EF=Φ/2 Inradius JK=\(\frac{\sqrt{25+10\sqrt5}}{10}\)=HJ CG=DG=Φ-1=1/Φ FH=\(\cos18 = \sqrt{\frac{5+\sqrt5}{8}}\) AF=FG=cos 54= \(\sqrt{\frac{5-\sqrt5}{8}}\) FJ=JK/Φ AG=\(2\cos 54 = \sqrt{\frac{5-\sqrt5}{2}}\) GJ=AJ/Φ CH=DH=BK=CK=1/2 GH=AF/Φ=\(\frac12\sqrt{5-2\sqrt5}\)