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)

$A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}} \simeq 7.694208843 t^2.$

An alternative formula is $\scriptstyle A\,=\,2.5dt$ where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry $\scriptstyle d\,=\,2t(\cos{54^\circ}\,+\,\cos{18^\circ})$.

Sides

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

Other websites

• Weisstein, Eric W., "Decagon" from MathWorld.
• Definition and properties of a decagon With interactive animation