**Spherical Triangles**

We discussed in the first session that geometry on the surface of a sphere differs from geometry on a plane. Of particular importance is the difference in trigonometry. Just as you can define a triangle on a planar surface, you can define a triangle on the surface of a sphere.

A planar triangle is the shape which connects three points by the shortest route (along straight lines). In the same way, a spherical triangle connects three points by the shortest route. Since we are now working on the surface of a sphere, the sides of a spherical triangle are no longer straight lines, but follow great circles, which we defined in the first session.

The complete definition of a spherical triangle is that it must fulfil all of the following conditions:

• each side is a part of a great circle,

• any two sides together are longer than the third side,

• each angle is less than 180°,

• the sum of the three angles is greater than 180°.

**Figure 1**: A spherical triangle on the surface of the Earth.