Decagon
A decagon is a shape with 10 sides and 10 corners (vertices), and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°.
Regular decagon
The area of a regular decagon is: (with t = edge length)
- [math]\displaystyle{ A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}} \simeq 7.694208843 t^2. }[/math]
An alternative formula is [math]\displaystyle{ A = 2.5dt }[/math] where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry [math]\displaystyle{ d = 2t(\cos{54^\circ} + \cos{18^\circ}) }[/math].
Sides
The side of a regular decagon inscribed in a unit circle is [math]\displaystyle{ \tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\phi} }[/math], where ϕ is the golden ratio, [math]\displaystyle{ \tfrac{1+\sqrt{5}}{2} }[/math].
Other websites
- Template:MathWorld
- Definition and properties of a decagon With interactive animation