area of a spherical triangle

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 tangentsPlanetmathPlanetmathPlanetmath to the incidentMathworldPlanetmathPlanetmath sides in the vertex tangent planeMathworldPlanetmath.

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

SABC=(A+B+C-π)R2. (1)

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

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

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 ABC is antipodal to ABC since it can be obtained by reflecting the original one through the center of the sphere. By symmetryMathworldPlanetmathPlanetmath, both triangles must have the same area.


For the proof of the above formula, the notion of a spherical diangle is helpful. As its name suggests, a diangle is formed by two great arcs that intersect in two points, which must lie on a diameterMathworldPlanetmath. Two diangles with vertices on the diameter AA are shown below.

At each vertex, these diangles form an angle of A. Similarly, we can form diangles with vertices on the diameters BB and CC respectively.

Note that these diangles cover the entire sphere while overlapping only on the triangles ABC and ABC. Hence, the total area of the sphere can be written as

Ssphere=2SAA+2SBB+2SCC-4SABC. (2)

Clearly, a diangle occupies an area that is proportional to the angle it forms. Since the area of the sphere ( is 4πR2, the area of a diangle of angle α must be 2αR2.

Hence, we can rewrite equation (2) as


which is the same as equation (1). ∎

Title area of a spherical triangle
Canonical name AreaOfASphericalTriangle
Date of creation 2013-03-22 14:21:38
Last modified on 2013-03-22 14:21:38
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 9
Author Mathprof (13753)
Entry type Theorem
Classification msc 51M25
Classification msc 51M04
Related topic AreaOfTheNSphere
Related topic Defect
Related topic SolidAngle
Related topic LimitingTriangle
Related topic SphericalTrigonometry