Wanted : for more formulas to find the area of a triangle?
Vectors: The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors. Let vectors $AB$ and $AC$ point respectively from $A$ to $B$ and from $A$ to $C$. The area of parallelogram ABDC is then $$\left|AB \times AC\right|$$ so that the area of a triangle is half of this, giving $$A_{\text{triangle}} = \frac{1}{2} |AB \times AC|.$$
Pick's Theorem: $$A_{\text{triangle}} = i + \frac{b}{2} - 1$$ where $i$ is the number of internal lattice points of a triangle and $b$ is the number of lattice points lying on the border of the triangle. As per mathlove: We require that all the triangle's vertices are on lattice points.
A two part paper by Marcus Baker (1849-1903) in vols. 1 and 2 of the Annals of Mathematics, readily available online, gives $110$ such formulae (warning: the Wikipedia article on triangles states that some of them are erroneous).
A collection of formulae for the area of a plane triangle] [Part 1], Annals of Mathematics (1) 1 #6 (January 1885), 134-138. JSTOR link google-books link archive.org link
A collection of formulae for the area of a plane triangle [Part 2], Annals of Mathematics (1) 2 #1 (September 1885), 11-18. JSTOR link google-books link archive.org link
Added as an edit since I can't comment. The links to these articles have been given above. While I'm at it, here is a systematic way to derive these formulae and even find your own new ones. Without loss of generality, one can assume that the vertices are $A=(0,0)$, $B=(1,0)$ and $C=(p,q)$. One can then spend a pleasant hour computing the metric quantities involved in the identities (side lengths, trigonometric functions of the angles, lengths of medians, angle bisectors, altitudes ....) in terms of $p$ and $q$. This reduces the problem to showing that an expression in these variables reduces to $\frac q 2$ or, after squaring, to $\frac{q^2} 4$. This can often be done by hand---in cases of emergency, one can use mathematica.
$s=pr$ where $p=\frac{a+b+c}{2}$ and $r$ is the radius of the inscribed circle.
$s=\sqrt{r\cdot r_a\cdot r_b\cdot r_c}$ where $r_a,r_b,r_c$ are the exradii of excircles.