Regular decagon
A regular decagon
Type	Regular polygon
Edges and vertices	10
Schläfli symbol	{10}, t{5}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D10), order 2×10
Internal angle (degrees)	144°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In geometry, a decagon (from the Greek δέκα déka and γωνία gonía, "ten angles") is a ten-sided polygon or 10-gon.^[1] 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°.^[1] Its Schläfli symbol is {10} ^[2] and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.

Decagons often appear in tilings with (partial) 5-fold symmetry. The images show an Islamic geometric pattern (15th century), an illustration in Kepler's Harmonices Mundi (1619) and a Penrose tiling.

Side length

The picture shows a regular decagon with side length ${\displaystyle a}$ and radius ${\displaystyle R}$ of the circumscribed circle.

• The triangle ${\displaystyle E_{10}E_{1}M}$ has two equally long legs with length ${\displaystyle R}$ and a base with length ${\displaystyle a}$
• The circle around ${\displaystyle E_{1}}$ with radius ${\displaystyle a}$ intersects E 10 M\,E_{10}M\,E_{10}P\,E_{10}}"> P E 10 {\displaystyle P\,E_{10}} is ${\displaystyle R-a}$.
• The isosceles triangles ${\displaystyle E_{10}E_{1}M\;}$ and ${\displaystyle PE_{10}E_{1}\;}$ have equal angles of 36° at the vertex, and so they are similar, hence: ${\displaystyle \;{\frac {a}{R}}={\frac {R-a}{a}}}$
• Multiplication with the denominators ${\displaystyle R,a>0}$ leads to the quadratic equation: ${\displaystyle \;a^{2}=R^{2}-aR\;}$
• This equation for the side length ${\displaystyle a\,}$ has one positive solution: ${\displaystyle \;a={\frac {R}{2}}(-1+{\sqrt {5}})}$

So the regular decagon can be constructed with ruler and compass.

Further conclusions

${\displaystyle \;R={\frac {2a}{{\sqrt {5}}-1}}={\frac {a}{2}}({\sqrt {5}}+1)\;}$ and the base height of ${\displaystyle \Delta \,E_{10}E_{1}M\,}$ (i.e. the length of ${\displaystyle M\,D}$) is ${\displaystyle h={\sqrt {R^{2}-(a/2)^{2}}}={\frac {a}{2}}{\sqrt {5+2{\sqrt {5}}}}\;}$ and the triangle has the area: ${\displaystyle A_{\mathrm{\Delta <!--\; \Delta \; -->}}=\frac{a}{2}\cdot <!--\; \cdot \; -->h=\frac{a^2}{4}\sqrt{5+2}}$Zdroj:https://en.wikipedia.org?pojem=Decagon
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.