The golden triangle, sometimes also called the sublime triangle, is an isosceles triangle such that the ratio of the hypotenuse to base is equal to the golden ratio ,
. From the above figure, this means
that the triangle has vertex angle equal to
(1)
or ,
and that the height
is related to the base
through
The inradius of a golden triangle is
(5)
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
(6)
for a pentagram and that the circumradius of a decagon
of side length
is
(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 ,
where
is the golden ratio .
See also Decagon ,
Golden Gnomon ,
Golden Ratio ,
Golden
Rectangle ,
Isosceles Triangle ,
Penrose
Tiles ,
Pentagram
Explore with Wolfram|Alpha
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
Subject classifications