Golden angle

In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the larger arc is the same as the ratio of the length of the larger arc to the full circumference of the circle.

Algebraically, let a+b be the circumference of a circle, divided into a longer arc of length a and a smaller arc of length b such that

${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}}$

The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately 137.5077640500378546463487...° OEIS: A096627 or in radians 2.39996322972865332... OEIS: A131988.

The name comes from the golden angle's connection to the golden ratio φ; the exact value of the golden angle is

${\displaystyle 360\left(1-{\frac {1}{\varphi }}\right)=360(2-\varphi )={\frac {360}{\varphi ^{2}}}=180(3-{\sqrt {5}}){\text{ degrees}}}$

or

${\displaystyle 2\pi \left(1-{\frac {1}{\varphi }}\right)=2\pi (2-\varphi )={\frac {2\pi }{\varphi ^{2}}}=\pi (3-{\sqrt {5}}){\text{ radians}},}$

where the equivalences follow from well-known algebraic properties of the golden ratio.

As its sine and cosine are transcendental numbers, the golden angle cannot be constructed using a straightedge and compass.^[1]