Geometry | Part IB, 2001

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

Suppose D\mathbf{D} is a regular dodecahedron, with centre the origin. Explain how each face of D\mathbf{D} gives rise to a spherical pentagon on the 2 -sphere S2S^{2}. For each such spherical pentagon, calculate its angles and area.

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