In geometry, a prism is a polyhedron with an n-sided polygonal base, another congruent parallel base (with the same rotational orientation), and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. Prisms are named for their base, so a prism with a pentagonal base is called a pentagonal prism. In regular prisms, all side faces (all squares) are at right angles to the bases. The number of prisms is infinite, approaching a circular base. The interior angles of the base faces is \(\frac{180(n-2)}{n}\), where n is the number of sides. This is really the only “difficult” …Read the Rest

Truncated Platonics

2015-02-02
Truncated Platonics

There are several Archimedean solids that are formed by the truncation (cutting off) of each corner of a Platonic solid. These can be shown in successive truncations from one shape to its dual. Original Truncation Rectification Bitruncation Birectification (dual)   Tetrahedron   Truncated Tetrahedron   …Read the Rest

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=\(\frac12\sqrt{5-2\sqrt5}\)  

Factoring numbers with square root terms

Quite often, when doing calculations on polyhedra, you will find yourself with complex equations like \(\frac{20+7\sqrt3}{1+2\sqrt3}\).

Is the top evenly divisible by the bottom? I certainly can’t tell just by looking at them.

There must be a method to factor the numerator.

sqr(A+B*sqr(C))

2013-07-27
sqr(A+B*sqr(C))

Lately, I have been doing a lot of math involving square roots of numbers added to square roots, in the form of \(\sqrt{A+B\sqrt{C}}\), this is called a “nested radical.” Normally, you would not be able to simplify any further, unless there was a common factor …Read the Rest

SketchUp Platonics

2012-01-24
SketchUp Platonics

I’ve been playing with Google’s Sketchup for a week or so and I had the thought of doing some polyhedra. It is often hard to mentally visualize a 3D object when looking at a 2D image, so I figured these would help. Here is the …Read the Rest

Octahedron

2011-11-07
Octahedron

The octahedron is a Platonic solid, which means it is made of all regular polygons for each face, being eight equilateral triangles arranged four at each vertex. With edge length of S, the surface area would be that of 8 equilateral triangles, \(8\cdot S^2\cdot \frac{\sqrt3}{4}=2\cdot …Read the Rest

Law of Sines and Cosines

The law of cosines relates the sides and angles of a triangle. \(a^2=b^2+c^2-2bc\cdot \cos\alpha \\ b^2=a^2+c^2-2ac\cdot \cos\beta \\ c^2=a^2+b^2-2ab\cdot \cos\gamma\) It can also be rearranged to: \(\large\alpha=\arccos\left(\frac{b^2+c^2-a^2}{2bc}\right) \\ \large\beta=\arccos\left(\frac{a^2+c^2-b^2}{2ac}\right) \\ \large\gamma=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)\) As long as all three sides or at least one side and two angles …Read the Rest

The Cube

2011-10-28
The Cube

The cube is probably the most recognized and best known of and 3D shape. Kids young enough not to even know how to talk will still know about cubes, they play with multicolored wooden blocks. Older kids may tackle the Rubik’s Cube puzzle or play …Read the Rest