Archive for January, 2015

Icosahedron

2015-01-29

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

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

A regular pentagon has the following dimensions: AB=BC=CD=DE=AE=BG=EG=1 Height AH=EK=$\frac12\cos 18 = \frac12\sin 72= \frac12\sqrt{5+2\sqrt5}$ AC=AD=BD=BE=CE=Φ Circumcenter AJ=$\sqrt{\frac{5+\sqrt5}{10}}$=BJ=CJ=DJ=EJ BF=EF=Φ/2 = cos 36 = sin 54 = $\frac{1+\sqrt5}{4}$ Inradius JK=HJ=$\sqrt{\frac{5+2\sqrt5}{20}}$=$\frac{AH}{\sqrt5}$ CG=DG= Φ-1 = 1/Φ = $\frac{\sqrt5-1}{2}$ FH= cos 18 =sin 72 = $\sqrt{\frac{5+\sqrt5}{8}}$ AF=FG= cos 54 …Read the Rest

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