Decagon

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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[change | change source]

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[change | change source]

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[change | change source]