Spherical Excess
the difference between the sum of the angles of a spherical triangle and 180°—that is, the amount by which the sum is greater than the sum of the angles of a plane triangle.
The sum of the angles of a triangle formed by three geodesies on a surface of positive curvature—that is, on a convex surface—is always greater than two right angles and is equal to
π + ʃʃ K dS
where K is the total curvature of the surface and dS is the element of area of the surface. The spherical excess of a spherical triangle formed by great circles on a sphere of radius R is
where S is the area of the triangle. In the case of small spherical triangles on the surface of the earth, the quantity ∊, in seconds of arc, is given by the formula
Here, a and b are the lengths of two sides of the triangle, and C is the angle between the two sides.
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