General Math

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

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

Phi, the Golden Ratio

Phi \((\Phi, \phi)\) is a Greek letter that mathematicians have assigned to a specific ratio or proportion, called the golden ratio, that most people find to be attractive in art, architecture, and nature. The golden ratio is illustrated as \(\frac{A}{B}=\frac{A+B}{A}\equiv\phi\). The only positive solution is …Read the Rest

Table of exact trigonometric functions

Since I noticed I had to keep looking some of these up, I place them here, just for reference, gathered from around the internet. Many of these formulas can be written in different ways, but I have simplified them as much as possible. Where, \(\phi=\frac{1+\sqrt{5}}{2}=1.6180339887498948482045868\ldots\), …Read the Rest