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

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. Monthly67, 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/.