A spherical triangle is formed by connecting three points on the surface of a sphere with great arcs; these three points do not lie on a great circle of the sphere. The measurement of an angle of a spherical triangle is intuitively obvious, since on a small scale the surface of a sphere looks flat. More precisely, the angle at each vertex is measured as the angle between the tangents to the incident sides in the vertex tangent plane.

Theorem. The area of a spherical triangle $ABC$ on a sphere of radius $R$ is

Incidentally, this formula shows that the sum of the angles of a spherical triangle must be greater than or equal to $\pi$, with equality holding in case the triangle has zero area.

Since the sphere is compact, there might be some ambiguity as to whether the area of the triangle or its complement is being considered. For the purposes of the above formula, we only consider triangles with each angle smaller than $\pi$.

An illustration of a spherical triangle formed by points $A$, $B$, and $C$ is shown below.

Note that by continuing the sides of the original triangle into full great circles, another spherical triangle is formed. The triangle $A^{{\prime}}B^{{\prime}}C^{{\prime}}$ is antipodal to $ABC$ since it can be obtained by reflecting the original one through the center of the sphere. By symmetry, both triangles must have the same area.