DECAGON
, a plane geometrical figure of ten sides and ten angles. When all the sides and angles are equal, it is a regular decagon, and may be inscribed in a circle; otherwise, not.
If the radius of a circle, or the side of the inscribed hexagon, be divided in extreme and mean proportion, the greater segment will be the side of a decagon inscribed in the same circle. And therefore, as the side of the decagon is to the radius, so is the radius to the sum of the two. Whence, if the radius of the circle be r, the side of the inscribed decagon will be (√(5-1))/2 X r.
If the side of a regular decagon be 1, its area will be 5/2 √(5+2√5)=7.6942088; therefore as 1 is to 7.6942088, so is the square of the side of any regular decagon, to the area of the same: so that, if s be the side of such a decagon, its area will be equal to 7.6942088s2. See Regular Figure.
To inscribe a decagon in a circle geometrically. See my Mensuration, prob. 35, pa. 25, 2d edit.
