regular decagon inscribed in circle∗
pahio†
2013-03-21 23:53:04
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 decagon, inscribed in a circle, is equal to
the greater part of the radius divided with the golden section.
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 angle O = 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 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.
A
36◦
r
36◦
s
36◦
s
72◦
s
72◦
r−s
B
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Privacy setting: h1i hTheoremi h51M04i
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O
∗ hRegularDecagonInscribedInCirclei
1
Note. (1) can be simplified to the quadratic equation
s2 +rs−r2 = 0
which yields the positive solution
s =
√
−1+ 5
r ≈ 0.618 r.
2
Cf. also the golden ratio.
2