Spherical Law of Cosines

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Draw a spherical triangle on the surface of the unit sphere with center at the origin . Let the sides (arcs) opposite the vertices have lengths , and , and let be the angle at vertex . The spherical law of cosines is then given by , with two analogs obtained by permutations.

Contributed by: Izidor Hafner (February 2017)
Open content licensed under CC BY-NC-SA



Let be the plane tangent to the sphere at , and let and . Then , , and . Express the length of in two ways using the usual planar law of cosines for the triangle in the plane :


With the triangle , the law of cosines gives


The two equalities give



[1] Wikipedia. "Spherical Law of Cosines." (Feb 22, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.