Editing Decagon (section)

=== Side length ===
[[File:01-Zehneck-Seitenlänge.svg|300px|right]]
The picture shows a regular decagon with side length <math>a</math> and radius <math>R</math> of the [[circumscribed circle]].

* The triangle <math>E_{10}E_1M</math> has two equally long legs with length <math>R</math> and a base with length <math>a</math>
* The circle around <math>E_1</math> with radius <math>a</math> intersects <math>]M\,E_{10}[</math> in a point <math>P</math> (not designated in the picture).
* Now the triangle <math>{E_{10}E_1P}\;</math> is an [[isosceles triangle]] with vertex <math>E_1</math> and with base angles <math>m\angle E_1 E_{10} P = m\angle E_{10} P E_1 = 72^\circ \;</math>.
* Therefore <math>m\angle P E_1 E_{10} = 180^\circ -2\cdot 72^\circ = 36^\circ \;</math>. So <math>\; m\angle M E_1 P = 72^\circ- 36^\circ = 36^\circ\;</math> and hence <math>\; E_1 M P\;</math> is also an isosceles triangle with vertex <math>P</math>. The length of its legs is <math>a</math>, so the length of <math>[P\,E_{10}]</math> is <math>R-a</math>.
* The isosceles triangles <math>E_{10} E_1 M\;</math> and <math>P E_{10} E_1\;</math> have equal angles of 36° at the vertex, and so they are [[Similarity (geometry)|similar]], hence: <math>\;\frac{a}{R}=\frac{R-a}{a}</math>
* Multiplication with the denominators <math>R,a >0</math> leads to the quadratic equation: <math>\;a^2=R^2-aR\;</math>
* This equation for the side length <math>a\,</math> has one positive solution: <math>\;a=\frac{R}{2}(-1+\sqrt{5})</math>
So the regular decagon can be constructed with ''[[Straightedge and compass construction|ruler and compass]]''.

;Further conclusions:
<math>\;R=\frac{2a}{\sqrt{5}-1}=\frac{a}{2}(\sqrt{5}+1)\;</math> and the base height of <math>\Delta\,E_{10} E_1 M\,</math> (i.e. the length of <math>[M\,D]</math>) is <math>h = \sqrt{R^2-(a/2)^2}=\frac{a}{2}\sqrt{5+2\sqrt{5}}\;</math> and the triangle has the area: <math>A_\Delta=\frac{a}{2}\cdot h = \frac{a^2}{4}\sqrt{5+2\sqrt{5}}</math>.