Help needed in understanding Heron's Formula So i just started learning Trigonometry seriously and something doesn't feel right to me, either I'm missing something or not but.
Lets assume we have a triangle

and there are two ways to find the area.
1 is using the standard $$A = \frac{1}{2}bh$$ and by using the example image above we get $A = 10625$
but if I use the other formula, in this case, Heron's Formula 
\begin{align*}
s & = \frac{a+b+c}{2}\\
A & = \sqrt{s(s-a)(s-b)(s-c)}
\end{align*}
the area becomes $A = 10620.09$.
They're both gravely close to each other, which has me thinking maybe I missed something.
So my question is, why are the areas different?
 A: You shouldn't consider the areas different, because the relative difference is smaller than $0.05\%$.
But you can a priori expect an error on the height to be up to $0.5\%$ as some number is truncated to integer. Hence without deeper error calculus, you can't be conclusive.
A: I just had time to construct the scene in GeoGebra:


*

*We add a point $B=(0,0)$ and a point $C=(170,0)$.

*Then we add a circle around $B$ with radius $r_1=169$
and a circle around $C$ with radius $r_2=137$.

*We use the intersection tool to determine the intersection points $A$ and $D$ (not shown) of the two circles.

*Then we use the polygon tool to create the triangle $ABC$.

*The perpendicular line tool then allows to pick the point $A$ and the line $BC$ to add the line through $A$, perpendicular to the line $BC$

*The intersection tool allows to define the intersection point $E$ by choosing that perpendicular line and the line $BC$.

*Finally we use the measurement tools to add the area of $ABC$, and the length of the segments $BA$, $CA$ and $AE$.



It shows that $AE=124.94$, confirming Paul, that the rounded value $125$ is responsible for the difference in calculated area.
Indeed
$$
\frac{170 \cdot 124.94}{2} = 10619.9
$$
and after digging out two more places
$$
\frac{170 \cdot 124.9422}{2} = 10620.087
$$
Appendix: Determining the height of the triangle:
We follow the geometric construction, calculating the intersection point $A=(x,y)$ of the two circles:
$$
x^2 + y^2 = 169^2 \\
(x - 170)^2 + y^2 = 137^2
$$
Subtracting the first from the second equation gives
$$
-340 x + 170^2 = 137^2 - 169^2 \iff \\
x = \frac{170^2 + 169^2 - 137^2}{340} = 113.8
$$
So we can e.g. use the first to calculate
$$
y = \pm \sqrt{169^2 - (113.8)^2} = \pm 124.94222664\dotsb
$$
