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# Decagon

[dek-uh-gon]
Regular decagon

Edges and vertices 10
Schläfli symbols {10}
t{5}
Coxeter–Dynkin diagrams >- Symmetry group Dihedral (D10)
Area
(with t=edge length)
$A = frac\left\{5\right\}\left\{2\right\}t^2 cot frac\left\{pi\right\}\left\{10\right\}$$= frac\left\{5t^2\right\}\left\{2\right\} sqrt\left\{5+2sqrt\left\{5\right\}\right\}$ $simeq 7.694208843 t^2.$
Internal angle
(degrees)
144°

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all angles equal to 144°, therefore making each angle of a regular decagon be 144°. Its Schläfli symbol is {10}. The area of a regular decagon of side length a is given by


begin{align} A & = frac{5}{2}a^2 cot frac{pi}{10} = frac{5a^2}{2} sqrt{5+2sqrt{5}}
`                 & simeq 7.694208843, a^2.`
end{align}

## Construction

A regular decagon is constructible with a compass and straightedge.

1. Complete steps 1 though 6 of constructing a pentagon.
2. Extend a line from each corner of the pentagon through the center of the circle made in step 1 of constructing a pentagon to the opposite side of that same circle.
3. The five corners of the pentagon constitute every other corner of the decagon. The remaining five corners of the decagon are those points where the lines of step 2 cross the original circle (but not a pentagon corner).