# regular decagon inscribed in circle

If a line segment  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.

Proof. A regular polygon can be inscribed in a circle (http://planetmath.org/RegularPolygonAndCircles). In the picture below, there is seen an isosceles central triangle $OAB$ of a regular decagon with the central angle  $O=360^{\circ}\!:\!10=36^{\circ}$; the base angles are  $(180^{\circ}\!-\!36^{\circ})\!:\!2=72^{\circ}$. 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 triangle  $OAB$. From these similar triangles  we obtain the proportion equation

 $\displaystyle r:s\,=\,s:(r\!-\!s),$ (1)

which shows that the side $s$ of the regular decagon is the central proportional of the radius $r$ of the circle and the difference  $r\!-\!s$.

 $s^{2}\!+\!rs\!-\!r^{2}=0$
 $s\;=\;\frac{-1\!+\!\sqrt{5}}{2}\,r\;\approx\;0.618\,r.$
Title regular decagon inscribed in circle RegularDecagonInscribedInCircle 2013-03-22 17:34:26 2013-03-22 17:34:26 pahio (2872) pahio (2872) 10 pahio (2872) Theorem msc 51M04 regular decagon RegularPolygonAndCircles HomogeneousEquation Pentagon  golden section