Golden Triangle

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

			(2)
			(3)
			(4)

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.