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Spherical Triangle

A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle (Harris and Stocker 1998). Let a spherical triangle have angles , , and (measured in radians at the vertices along the surface of the sphere) and let the sphere on which the spherical triangle sits have radius . Then the surface area of the spherical triangle iswhere is called the spherical excess, with in the degenerate case of a planar triangle.The sum of the angles of a spherical triangle is between and radians ( and ; Zwillinger 1995, p. 469). The amount by which it exceeds is called the spherical excess and is denoted or , the latter of which can cause confusion since it also can refer to the surface area of a spherical triangle. The difference between radians () and the sum of the side arc lengths , ,..

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