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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,

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

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)

b+l={\textstyle{1\over 2}}a(\sqrt{5}+1)
\end{displaymath} (3)

{b+a\over a} = {\sqrt{5}+1\over 2}=\phi.
\end{displaymath} (4)

See also Decagon, Golden Ratio, Isosceles Triangle, Pentagram


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

© 1996-9 Eric W. Weisstein