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Exercise 3.1 — 15 pts. A spherical polygon is a region on the sphere bordered by great
circles. In this exercise you will show that the sum ˝i θi of the exterior angles θi of a spherical
polygon is 2π − A, where A is the area enclosed by the spherical polygon.

(a) 5 pts. Show that the area A of a spherical triangle on the unit sphere with interior
angles α1;α2;α3 is

A = α1 + α2 + α3 − π: (1)

Hint Consider the areas A1;A2;A3 of the three spherical lunes.

(b) 5 pts. Show that the area of a spherical n-gon with interior angles β1; : : : ; βn is

Hint Partition the polygon into triangles.

(c) 5 pts. Conclude that the area of a spherical n-gon with exterior angles θ1; : : : ; θn is


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