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.

Theorem. The side of the regular (http://planetmath.org/RegularPolygon) decagon (http://planetmath.org/Polygon), inscribed 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 (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

$r:s=s:(r-s),$(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 $r-s$.

Note. (1) can be simplified to the quadratic equation (http://planetmath.org/QuadraticFormula)

$$s^2+rs-r^2=0$$

which yields the positive solution

$$s=\frac{-1+\sqrt{5}}{2}r\approx \mathrm{\hspace{0.33em}0.618}r.$$

Cf. also the golden ratio.