Spherical trigonometry is the study of curved triangles, triangles drawn on the surface of a sphere. The subject is practical, for example, because we live on a sphere. The subject has numerous elegant and unexpected theorems. We give a few below.

The diagram shows the spherical triangle with vertices *A*, *B*, and *C*. The angles at each vertex are denoted with Greek letters α, β, and γ. The arcs forming the sides of the triangle are labeled by the lower-case form of the letter labeling the opposite vertex.

## Basic properties

On the plane, the sum of the interior angles of any triangle is exactly 180°. On a sphere, however, the corresponding sum is always greater than 180° but also less than 540°. That is, 180° < α + β + γ < 540° in the diagram above. The positive quantity *E* = α + β + γ – 180° is called the **spherical excess** of the triangle.

Since the sides of a spherical triangle are arcs, they can be described as angles, and so we have two kinds of angles:

- The angles at the vertices of the triangle, formed by the great circles intersecting at the vertices and denoted by Greek letters.
- The sides of the triangle, measured by the angle formed by the lines connecting the vertices to the center of the sphere and denoted by lower-case Roman letters.

The second kind of angle is most interesting. In contrast to plane trigonometry, the sides of a spherical triangle are themselves are angles, and so we can take sines and cosines etc. of the *sides* as well as the vertex angles.

## Right spherical triangles

For this section, assume the angle γ = 90°, i.e. we have a spherical right triangle. Then the following identities hold.

- sin
*a*= sin α sin*c*= tan*b*cot β - sin
*b*= sin β sin*c*= tan*a*cot α - cos α = cos
*a*sin β = tan*b*cot*c* - cos β = cos
*b*sin α = tan*a*cot*c* - cos
*c*= cot α cot β = cos*a*cos*b*

Napier’s rule is a mnemonic for memorizing the above identities.

## General spherical triangle

For this section we drop the assumption that γ = 90°. Many identities hold. Here are a few examples.

### Law of sines

sin α / sin *a* = sin β / sin *b* = sin γ / sin *c*

### Law of cosines

cos *a* = cos *b* cos *c* + sin *b* sin *c* cos α

cos α = -cos β cos γ + sin β sin γ cos *a*

### Tangents

Let *s* = (*a* + *b* + *c*)/2 and let σ = (α + β + γ)/2. The following formulas can used to solve for a vertex angle from knowing the side arcs or solve for a side arc from knowing the vertex angle.

tan (α/2) = √( sin(*s*–*b*) sin(*s*–*c*) / sin *s* sin(*s*–*a*) )

tan (*a* /2) = √( -cos(σ) cos(σ-α) / cos(σ-β) cos(σ-γ) )

### Area

Let *R* be the radius of the sphere on which a triangle resides. If angles are measured in radians, the area of a triangle is simply *R*^{2}*E* where *E* is the spherical excess, defined above. In degrees the formula for area is π*R*^{2}*E*/180.

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