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)
|
where is the circumradius,
is the inradius,
and
is the semiperimeter.
A particularly beautiful formula for is Heron's formula
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(8)
|
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)
|
where is the determinant and
is a two-dimensional
cross product.
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)
|
gives the particularly pretty form
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(14)
|
For additional formulas, see Beyer and Baker, 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
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(15)
|
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)
|
If the triangle is embedded in three-dimensional space with the coordinates of the vertices given by , then
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(18)
|
This can be written in the simple concise form
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(19)
|
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(20)
|
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)
|
where is the area of the reference triangle. For arbitrary trilinears, the equation then becomes
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(22)
|