The area 
(sometimes also denoted 
)
of a triangle 
with side lengths 
,
, 
and corresponding angles 
, 
,
and 
is given by
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(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
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(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.,
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(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),
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(11)
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(12)
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(13)
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gives the particularly pretty form
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(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
![Delta=2R^2|sin(1/2theta_1)sin(1/2theta_2)sin[1/2(theta_1-theta_2)]|.](/images/equations/TriangleArea/NumberedEquation3.svg) |
(15)
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The (signed) area of a planar triangle specified by its vertices 
for 
,
2, 3 is given by
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(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
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(18)
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This can be written in the simple concise form
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(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
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(21)
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where 
is the area of the reference triangle (Kimberling 1998, p. 35). For arbitrary
trilinears, the equation then becomes
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(22)
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