Angles of Spherical Polygons

When filled in, this will be a table of data comparing interior angle sums on the sphere with corresponding angle sums in the Euclidean plane. Where possible, the area will also be tabulated.

Note: If a polygon is part of a tessellation of the sphere, you can figure out angles by noting that the angle sum of vertices at a single point still add up to 360 degrees.

		Measured in degrees			Measured in radians
Case	Polygon type	Angle sum on sphere	Expected Euclidean angle sum	Diff. of angle sums	Angle sum on sphere	Expected Euclidean angle sum	Diff. of angle sums	Area as fraction of total sphere area
Octahedron fact	Triangle
Cube Face	Quadri-lateral
Tetrahedron face	Triangle
Icosahedron face	Triangle
Dodecahedron face	Pentagon
Half-lune with 60-degree angle	Triangle
Half-lune with 45-degree angle	Triangle
"1/8" of cube face	Triangle

On graph paper, plot area against the difference (using either degree measurement) and look for a relationship.