This post will look at the triangle behind North Carolina’s Research Triangle using Mathematica’s geographic functions.

## Spherical Triangles

A spherical triangle is a triangle drawn on the surface of a sphere. It has three vertices, given by points on the sphere, and three sides. The sides of the triangle are portions of great circles running between two vertices. A great circle is a circle of maximum radius, a circle with the same center as the sphere.

An interesting aspect of spherical geometry is that both the sides and angles of a spherical triangle are angles. Because the sides of a spherical triangle are arcs, they have angular measure, the angle formed by connecting each vertex to the center of the sphere. The arc length of a side is its angular measure times the radius of the sphere.

Denote the three vertices by *A*, *B*, and *C*. Denote the side opposite *A* by *a*, etc. Denote the angles at *A*, *B*, and *C*by α, β, and γ respectively.

## Research Triangle

Research Triangle is a (spherical!) triangle with vertices formed by Duke University, North Carolina State University, and University of North Carolina at Chapel Hill.

(That was the origin of the name, though it’s now applied more loosely to the general area around these three universities.)

We’ll take as our vertices

*A*= UNC Chapel Hill (35.9046 N, 79.0468 W)*B*= Duke University in Durham (36.0011 N, 78.9389 W),*C*= NCSU in Raleigh (35.7872 N, 78.6705 W)