Wanted : for more formulas to find the area of a triangle? I know some formulas to find a triangle's area, like the ones below.


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*Is there any reference containing most triangle area formulas?  

*If you know more, please add them as an answer 


$$s=\sqrt{p(p-a)(p-b)(p-c)} ,p=\frac{a+b+c}{2}\\s=\frac{h_a*a}{2}\\s=\frac{1}{2}bc\sin(A)\\s=2R^2\sin A \sin B \sin C$$ 
Another symmetrical form is given by :$$(4s)^2=\begin{bmatrix}
a^2 &  b^2 & c^2  
\end{bmatrix}\begin{bmatrix}
-1 & 1  & 1\\ 
1 &  -1 & 1\\ 
1 & 1 & -1
\end{bmatrix} \begin{bmatrix}
a^2\\ 
b^2\\ 
c^2
\end{bmatrix}$$

Expressing the side lengths $a$, $b$ & $c$ in terms of the radii $a'$, $b'$ & $c'$ of the mutually tangent circles centered on the triangle's vertices (which define the Soddy circles)
$$a=b'+c'\\b=a'+c'\\c=a'+b'$$gives the paticularly pretty form $$s=\sqrt{a'b'c'(a'+b'+c')}$$
If the triangle is embedded in three dimensional space with the coordinates of the vertices given by $(x_i,y_i,z_i)$ then $$s=\frac{1}{2}\sqrt{\begin{vmatrix}
y_1 &z_1  &1 \\ 
 y_2&z_2  &1 \\ 
y_3 &z_3  &1 
\end{vmatrix}^2+\begin{vmatrix}
z_1 &x_1  &1 \\ 
 z_2&x_2  &1 \\ 
z_3 &x_3  &1 
\end{vmatrix}^2+\begin{vmatrix}
x_1 &y_1  &1 \\ 
 x_2&y_2  &1 \\ 
x_3 &y_3  &1 
\end{vmatrix}^2}$$
When we have 2-d coordinate $$ s=\frac{1}{2}\begin{vmatrix}
x_a &y_a  &1 \\ 
x_b &y_b  &1 \\ 
x_c &y_c  & 1
\end{vmatrix}$$

In the above figure, let the circumcircle passing through a triangle's vertices have radius $R$, and denote the central angles from the first point to the second $q$, and to the third point by $p$ then the area of the triangle is given by:
$$ s=2R^2|\sin(\frac{p}{2})\sin(\frac{q}{2})\sin(\frac{p-q}{2})|$$

I have a little more:)
$$
S = 4R^2(\sin A + \sin B + \sin C)\sin\frac A2\sin\frac B2\sin\frac C2\\
S = \frac{r^2}{4}\frac{\sin A + \sin B + \sin C}{\sin\dfrac A2\sin\dfrac B2\sin\dfrac C2}\\
S = r^2\left(\cot\frac A2+\cot\frac B2 + \cot\frac C2\right)\\
S = r^2\cot\frac A2 \cot\frac B2  \cot\frac C2\\
S = 2p^2 \frac{\sin A\sin B\sin C}{(\sin A+\sin B+\sin C)^2}\\
S = 4p^2 \frac{\sin\dfrac A2 \sin\dfrac B2 \sin\dfrac C2}{\sin A+\sin B+\sin C}
$$
