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# Triangle Area

The area (sometimes also denoted ) of a triangle with side lengths , , and corresponding angles , , and is given by

 (1) (2) (3) (4) (5) (6) (7)

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)

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) (10)

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) (12) (13)

gives the particularly pretty form

 (14)

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)

The (signed) area of a planar triangle specified by its vertices for , 2, 3 is given by

 (16) (17)

If the triangle is embedded in three-dimensional space with the coordinates of the vertices given by , then

 (18)

This can be written in the simple concise form

 (19) (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

 (21)

where is the area of the reference triangle (Kimberling 1998, p. 35). For arbitrary trilinears, the equation then becomes

 (22)

Area, Heron's Formula, Point-Line Distance--3-Dimensional, Polygon Area, Quadrilateral, Triangle

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## References

Baker, M. "A Collection of Formulæ for the Area of a Plane Triangle." Ann. Math. 1, 134-138, 1884.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 123-124, 1987.Ivanoff, V. F. "Solution to Problem E1376: Bretschneider's Formula." Amer. Math. Monthly 67, 291-292, 1960.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Triangle Area

## Cite this as:

Weisstein, Eric W. "Triangle Area." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangleArea.html