Decagon

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

In geometry, a decagon is a ten-sided polygon or 10-gon.

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.

The area of a regular decagon of side length a is given by:

In terms of the apothem r (see also inscribed figure), the area is:

In terms of the circumradius R, the area is:

An alternative formula is $A=2.5da$ $A=2.5da$ where d is the distance between parallel sides, or the height when the decagon stands on one side as base, or the diameter of the decagon's inscribed circle. By simple trigonometry,

and it can be written algebraically as

The side of a regular decagon inscribed in a unit circle is ${\tfrac {-1+{\sqrt {5}}}{2}}={\tfrac {1}{\phi }}$ ${\tfrac {-1+{\sqrt {5}}}{2}}={\tfrac {1}{\phi }}$ , where ϕ is the golden ratio, ${\tfrac {1+{\sqrt {5}}}{2}}$ ${\tfrac {1+{\sqrt {5}}}{2}}$ .

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