Spherical Triangle
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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
![Delta=R^2[(A+B+C)-pi]=R^2E,](/images/equations/SphericalTriangle/NumberedEquation1.gif) |
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.
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5-ary Lyndon words of length 12
