<s>[INST] 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.


*

*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:
$$ s=2R^2|\sin(\frac{p}{2})\sin(\frac{q}{2})\sin(\frac{p-q}{2})|$$
 [/INST] $\require{cancel}$
Some errors
spotted in 
Baker's papers, referenced in
the accepted answer.
Expression N20 
is definitely wrong (even the dimension): 
\begin{align} 
20.\ &
\cancel{\color{blue}{\frac{R\,r}{\beta_a\beta_b\beta_c}\,
\left(\frac1a+\frac1b\right)
\left(\frac1b+\frac1c\right)
\left(\frac1c+\frac1a\right)}}
,
\end{align} 
and most likely,
the term $\beta_a\beta_b\beta_c$
must be in the numerator
and a constant $\tfrac12$ is missing, 
so the correct form
should be
\begin{align} 
20.\ &
\tfrac12\,{R\,r}{\beta_a\beta_b\beta_c}\,
\left(\frac1a+\frac1b\right)
\left(\frac1b+\frac1c\right)
\left(\frac1c+\frac1a\right)
.
\end{align} 
In the second
Baker's paper,
 expressions N63, N80 
are wrong : 
\begin{align}
63.\ &
\cancel{\color{blue}{R^2\sin 2A (1+\cos C)}}
\end{align}
\begin{align}
80.\ &
\cancel{\color{blue}{a\,(-\beta_a\,\sin \tfrac12\,A+\beta_b\,\sin \tfrac12\,B+\beta_c\,\sin \tfrac12\,C)}}
\\
\quad&=
\cancel{\color{blue}{2\,s\,(\beta_a\,\sin \tfrac12\,A+\beta_b\,\sin \tfrac12\,B+\beta_c\,\sin \tfrac12\,C)}}
.
\end{align}
Also, in N94 the two first expressions are correct:
\begin{align}
94.\ &
r^2\cot\tfrac12\,A+2\,R\,r\,\sin A
\\
&=
r_a^2\cot\tfrac12\,A-2\,R\,r_a\,\sin A
,
\end{align}
but there are obvious typos in the third and fourth. 
They should be
\begin{align}
&=r_b^2\cot\tfrac12\,B-2\,R\,r_b\,\sin B
\\
&=r_c^2\cot\tfrac12\,C-2\,R\,r_c\,\sin C
.
\end{align}
</s>