decagon

In geometry, a decagon (from the Greek δέκα déka and γωνία gonía, "ten angles") is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°. ==Regular decagon== A regular decagon has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is {10} and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.

=== Side length === 300px|right The picture shows a regular decagon with side length a and radius R of the circumscribed circle. The triangle E_{10}E_1M has two equally long legs with length R and a base with length a The circle around E_1 with radius a intersects ]M\,E_{10}[ in a point P (not designated in the picture). Now the triangle {E_{10}E_1P}\; is an isosceles triangle with vertex E_1 and with base angles m\angle E_1 E_{10} P = m\angle E_{10} P E_1 = 72^\circ \;. Therefore m\angle P E_1 E_{10} = 180^\circ -2\cdot 72^\circ = 36^\circ \;. So \; m\angle M E_1 P = 72^\circ- 36^\circ = 36^\circ\; and hence \; E_1 M P\; is also an isosceles triangle with vertex P. The length of its legs is a, so the length of [P\,E_{10}] is R-a. The isosceles triangles E_{10} E_1 M\; and P E_{10} E_1\; have equal angles of 36° at the vertex, and so they are similar, hence: \;\frac{a}{R}=\frac{R-a}{a} Multiplication with the denominators R,a >0 leads to the quadratic equation: \;a^2=R^2-aR\; This equation for the side length a\, has one positive solution: \;a=\frac{R}{2}(-1+\sqrt{5}) So the regular decagon can be constructed with ruler and compass. Further conclusions: \;R=\frac{2a}{\sqrt{5}-1}=\frac{a}{2}(\sqrt{5}+1)\; and the base height of \Delta\,E_{10} E_1 M\, (i.e. the length of [M\,D]) is h = \sqrt{R^2-(a/2)^2}=\frac{a}{2}\sqrt{5+2\sqrt{5}}\; and the triangle has the area: A_\Delta=\frac{a}{2}\cdot h = \frac{a^2}{4}\sqrt{5+2\sqrt{5}}.