3.I.4B

State and prove the Gauss-Bonnet theorem for the area of a spherical triangle.

Suppose $\mathbf{D}$ is a regular dodecahedron, with centre the origin. Explain how each face of $\mathbf{D}$ gives rise to a spherical pentagon on the 2 -sphere $S^{2}$. For each such spherical pentagon, calculate its angles and area.

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