Sum of angles of a triangle to zero. For a spherical triangle, the sum of the angles is greater than 180° and can be up to 540°. Specifically, the sum of the angles is where "f" is the fraction of the sphere's area which is enclosed by the triangle. Note that spherical geometry does not satisfy several of Euclid's axioms (including the parallel postulate.) Angles between adjacent sides of a triangle are referred to as "interior" angles in Euclidean and other geometries. "Exterior" angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem. One can also consider the
Pentagon theorem and the left-hand triangle as: The side "s" is therefore: a well-established result. Consequently, this construction of the pentagon is valid. "See main article": Carlyle circle The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation. This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows: Steps 6–8 are equivalent to the following version, shown in the animation: This follows quickly from the knowledge that twice the sine of 18 degrees is the reciprocal golden ratio, which we know geometrically from the triangle with angles of