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Golden Triangle


GoldenTriangle

The golden triangle, sometimes also called the sublime triangle, is an isosceles triangle such that the ratio of the hypotenuse a to base b is equal to the golden ratio, a/b=phi. From the above figure, this means that the triangle has vertex angle equal to

 theta=2sin^(-1)(b/(2a))=2sin^(-1)(1/(2phi))=1/5pi,
(1)

or 36 degrees, and that the height h is related to the base b through

h=sqrt((bphi)^2-(1/2b)^2)
(2)
=bsqrt(phi^2-1/4)
(3)
=1/2bsqrt(5+2sqrt(5)).
(4)

The inradius of a golden triangle is

 r=1/2bsqrt(5-2sqrt(5)).
(5)
GoldenTriangleFigures

The triangles at the tips of a pentagram (left figure) and obtained by dividing a decagon by connecting opposite vertices (right figure) are golden triangles. This follows from the fact that

 a/b=phi
(6)

for a pentagram and that the circumradius R of a decagon of side length s is

 R=phis.
(7)

Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2002, p. 79).

Successive points dividing a golden triangle into golden gnomons and triangles lie on a logarithmic spiral (Livio 2002, p. 119).

Kimberling (1991) defines a second type of golden triangle in which the ratio of angles is phi:1, where phi is the golden ratio.


See also

Decagon, Golden Gnomon, Golden Ratio, Golden Rectangle, Isosceles Triangle, Penrose Tiles, Pentagram

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References

Bicknell, M.; and Hoggatt, V. E. Jr. "Golden Triangles, Rectangles, and Cuboids." Fib. Quart. 7, 73-91, 1969.Hoggatt, V. E. Jr. The Fibonacci and Lucas Numbers. Boston, MA: Houghton Mifflin, 1969.Kimberling, C. "A New Kind of Golden Triangle." In Applications of Fibonacci Numbers: Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications,' Wake Forest University (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 171-176, 1991.Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 78-79, 2002.Pappas, T. "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989.Schoen, R. "The Fibonacci Sequence in Successive Partitions of a Golden Triangle." Fib. Quart. 20, 159-163, 1982.Wang, S. C. "The Sign of the Devil... and the Sine of the Devil." J. Rec. Math. 26, 201-205, 1994.

Cite this as:

Weisstein, Eric W. "Golden Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoldenTriangle.html

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