The area Delta (sometimes also denoted sigma) of a triangle DeltaABC with side lengths a, b, c and corresponding angles A, B, and C is given by

Delta=1/2bcsinA
(1)
=1/2casinB
(2)
=1/2absinC
(3)
=1/4sqrt((a+b+c)(b+c-a)(c+a-b)(a+b-c))
(4)
=1/4sqrt(2b^2c^2+2c^2a^2+2a^2b^2-a^4-b^4-c^4)
(5)
=(abc)/(4R)
(6)
=rs,
(7)

where R is the circumradius, r is the inradius, and s=(a+b+c)/2 is the semiperimeter.

A particularly beautiful formula for Delta is Heron's formula

 Delta=sqrt(s(s-a)(s-b)(s-c)).
(8)

If a triangle is specified by vectors u and v originating at one vertex, then the area is given by half that of the corresponding parallelogram, i.e.,

A=1/2|det(uv)|
(9)
=1/2|uxv|,
(10)

where det(A) is the determinant and uxv is a two-dimensional cross product.

Expressing the side lengths a, b, and c in terms of the radii a^', b^', and c^' of the mutually tangent circles centered on the triangle vertices (which define the Soddy circles),

a=b^'+c^'
(11)
b=a^'+c^'
(12)
c=a^'+b^',
(13)

gives the particularly pretty form

 Delta=sqrt(a^'b^'c^'(a^'+b^'+c^')).
(14)

For additional formulas, see Beyer and Baker, who gives 110 formulas for the area of a triangle.

TriangleInscribing

In the above figure, let the circumcircle passing through a triangle's polygon vertices have radius R, and denote the central angles from the first point to the second theta_1, and to the third point by theta_2. 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)]|.
(15)

The (signed) area of a planar triangle specified by its vertices v_i=(x_i,y_i) for i=1, 2, 3 is given by

Delta=1/(2!)|x_1 y_1 1; x_2 y_2 1; x_3 y_3 1|
(16)
=1/2(-x_2y_1+x_3y_1+x_1y_2-x_3y_2-x_1y_3+x_2y_3).
(17)

If the triangle is embedded in three-dimensional space with the coordinates of the vertices given by v_i=(x_i,y_i,z_i), then

 Delta=1/2sqrt(|y_1 z_1 1; y_2 z_2 1; y_3 z_3 1|^2+|z_1 x_1 1; z_2 x_2 1; z_3 x_3 1|^2+|x_1 y_1 1; x_2 y_2 1; x_3 y_3 1|^2).
(18)

This can be written in the simple concise form

Delta=1/2|(x_2-x_1)x(x_1-x_3)|
(19)
=1/2|(x_3-x_1)x(x_3-x_2)|,
(20)

where AxB denotes the cross product.

If the vertices of the triangle are specified in exact trilinear coordinates as a_i^':b_i^':c_i^', then the area of the triangle is

 Delta^'=(abc)/(8Delta^2)|a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3|,
(21)

where Delta is the area of the reference triangle. For arbitrary trilinears, the equation then becomes

 Delta^'=(abcDelta)/((aalpha_1+bbeta_1+cgamma_1)(aalpha_2+bbeta_2+cgamma_2)(aalpha_3+bbeta_3+cgamma_3))|alpha_1 beta_1 gamma_1; alpha_2 beta_2 gamma_2; alpha_3 beta_3 gamma_3|.
(22)