# area of a spherical triangle

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

 $S_{ABC}=(\angle A+\angle B+\angle C-\pi)R^{2}.$ (1)

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.

###### Proof.

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 diameter  . Two diangles with vertices on the diameter $AA^{\prime}$ are shown below. At each vertex, these diangles form an angle of $\angle A$. Similarly, we can form diangles with vertices on the diameters $BB^{\prime}$ and $CC^{\prime}$ respectively.  Note that these diangles cover the entire sphere while overlapping only on the triangles $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$. Hence, the total area of the sphere can be written as

 $S_{\mathrm{sphere}}=2S_{AA^{\prime}}+2S_{BB^{\prime}}+2S_{CC^{\prime}}-4S_{ABC}.$ (2)

Clearly, a diangle occupies an area that is proportional to the angle it forms. Since the area of the sphere (http://planetmath.org/AreaOfTheNSphere) is $4\pi R^{2}$, the area of a diangle of angle $\alpha$ must be $2\alpha R^{2}$.

Hence, we can rewrite equation (2) as

 $\displaystyle 4\pi R^{2}=2R^{2}(2\angle A+2\angle B+2\angle C)-4S_{ABC},$ $\displaystyle\therefore~{}S_{ABC}=(\angle A+\angle B+\angle C-\pi)R^{2},$

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