The area (sometimes also denoted ) of a triangle with side lengths , , and corresponding angles , , and is given by
(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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where is the circumradius, is the inradius, and is the semiperimeter (Kimberling 1998, p. 35; Trott 2004, p. 65).
A particularly beautiful formula for is Heron's formula
(8)
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If a triangle is specified by vectors and originating at one vertex, then the area is given by half that of the corresponding parallelogram, i.e.,
(9)
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(10)
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where is the determinant and is a two-dimensional cross product (Ivanoff 1960).
Expressing the side lengths , , and in terms of the radii , , and of the mutually tangent circles centered on the triangle vertices (which define the Soddy circles),
(11)
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(12)
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(13)
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gives the particularly pretty form
(14)
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For additional formulas, see Beyer (1987) and Baker (1884), who gives 110 formulas for the area of a triangle.
In the above figure, let the circumcircle passing through a triangle's polygon vertices have radius , and denote the central angles from the first point to the second , and to the third point by . Then the area of the triangle is given by
(15)
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The (signed) area of a planar triangle specified by its vertices for , 2, 3 is given by
(16)
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(17)
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If the triangle is embedded in three-dimensional space with the coordinates of the vertices given by , then
(18)
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This can be written in the simple concise form
(19)
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(20)
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where denotes the cross product.
If the vertices of the triangle are specified in exact trilinear coordinates as , then the area of the triangle is
(21)
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where is the area of the reference triangle (Kimberling 1998, p. 35). For arbitrary trilinears, the equation then becomes
(22)
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