Golden Triangle

$\begin{figure}\begin{center}\BoxedEPSF{GoldenTriangle.epsf scaled 800}\end{center}\end{figure}$

An Isosceles Triangle with Vertex angles 36°. Such Triangles occur in the Pentagram and Decagon. The legs are in a Golden Ratio to the base. For such a Triangle,

$\begin{displaymath} \sin(18^\circ) = \sin({\textstyle{1\over 10}}\pi) = {{{\textstyle{1\over 2}}b}\over l} \end{displaymath}$

(1)

$\begin{displaymath} b=2a\sin({\textstyle{1\over 10}}\pi) = 2a {\sqrt{5}-1\over 4} = {\textstyle{1\over 2}}a(\sqrt{5}-1) \end{displaymath}$

(2)

$\begin{displaymath} b+l={\textstyle{1\over 2}}a(\sqrt{5}+1) \end{displaymath}$

(3)

$\begin{displaymath} {b+a\over a} = {\sqrt{5}+1\over 2}=\phi. \end{displaymath}$

(4)

See also Decagon, Golden Ratio, Isosceles Triangle, Pentagram

References

Pappas, T. ``The Pentagon, the Pentagram & the Golden Triangle.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989.