regular decagon inscribed in circle

If a line segmentMathworldPlanetmath has been divided into two parts such that the greater part is the central proportional of the whole segment and the smaller part, then one has performed the golden section (Latin sectio aurea) of the line segment.

TheoremMathworldPlanetmath. The side of the regularPlanetmathPlanetmathPlanetmath ( decagon (, inscribedMathworldPlanetmath in a circle, is equal to the greater part of the radius divided with the .

Proof. A regular polygon can be inscribed in a circle ( In the picture below, there is seen an isosceles central triangle OAB of a regular decagon with the central angleMathworldPlanetmathO=360:10=36; the base angles are  (180-36):2=72. One of the base angles is halved with the line AC, when one gets a smaller isosceles triangle ABC with equal angles as in the triangleMathworldPlanetmath OAB. From these similar trianglesMathworldPlanetmath we obtain the proportion equation

r:s=s:(r-s), (1)

which shows that the side s of the regular decagon is the central proportional of the radius r of the circle and the differencePlanetmathPlanetmath r-s.


Note. (1) can be simplified to the quadratic equation (


which yields the positive solution

s=-1+52r 0.618r.

Cf. also the golden ratio.

Title regular decagon inscribed in circle
Canonical name RegularDecagonInscribedInCircle
Date of creation 2013-03-22 17:34:26
Last modified on 2013-03-22 17:34:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 51M04
Synonym regular decagon
Related topic RegularPolygonAndCircles
Related topic HomogeneousEquation
Related topic PentagonMathworldPlanetmath
Defines golden section