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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 is
where  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  ,  , and  is called the spherical defect and is denoted
 or  .
On any sphere, if three connecting arcs are drawn, two triangles are created. If each triangle takes up one hemisphere, then they are equal in size, but in general
there will be one larger and one smaller. Any spherical triangle can therefore be
considered both an inner and outer triangle, with the inner triangle usually being
assumed. The sum of the angles of an outer spherical triangle is between  and  radians.
The study of angles and distances of figures on a sphere is known as spherical trigonometry.
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, p. 79, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton,
FL: CRC Press, pp. 131 and 147-150, 1987.
Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). "The Spherical Triangle." §12.2 in VNR Concise Encyclopedia of Mathematics, 2nd ed. New York:
Van Nostrand Reinhold, pp. 262-272, 1989.
Green, R. M. Textbook on Spherical Astronomy, 6th ed. Cambridge, England:
Cambridge University Press, 1985.
Harris, J. W. and Stocker, H. "General Spherical Triangle." §4.9.1 in Handbook of Mathematics and Computational Science. New
York: Springer-Verlag, pp. 108-109, 1998.
Hartle, J. B. Gravity: An Introduction to Einstein's General Relativity.
San Francisco: Addison-Wesley, p. 18, 2003.
Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England:
Cambridge University Press, 1960.
Zwillinger, D. (Ed.). "Spherical Geometry and Trigonometry." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton,
FL: CRC Press, pp. 468-471, 1995.
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