(This works for all triangles)
The area ($A$) of a right triangle, can be given by the inner radius ($r$) multiplied by the semi perimeter ($s$), where $r = \frac12(a + b- c)$ and $s = \frac12(a + b + c)$.
$$A = \tfrac14\left(a+ b - c\right)\left(a + b + c\right)$$
Proof through Pythagorean Theorem:
$$\begin{align} \tfrac12ab &= \tfrac14\left(a+ b - c)(a + b + c\right) \\[8pt] 2ab &= (a+ b - c)(a + b + c)\\[4pt] &= \left(a+ b)^2 - c^2\right)\\[4pt] &= \left(a^2 + 2ab + b^2 - c^2\right)\\[4pt] 0 &= a^2 + b^2 - c^2\\[4pt] c^2 &= a^2 + b^2 \end{align}$$
If we are dealing with Pythagorean triples it pays to look at Euclid' formula, shown here as
$$A=m^2-k^2\qquad B=2mk\qquad C=m^2+k^2$$
Heron's formula is great for finding area of an arbitrary triangle, but there is no need for it if we are dealing with Pythagorean triples. Instead, we can straight to the area/perimeter ratio. These occur in multiples of 1/2 starting with triangle $(3,4,5)$. To find triples with any such ration (here R-1), we may begin by solving equations for $k$ and testing a define range of m-values to see which, if any, yield integers.
$$\text{Using $F(m,k)$: }\qquad R=\frac{area}{perimeter}=\frac{AB/2P}=\frac{2mk(m^2-k^2)}{2(2m^2+2mk)}=\frac{mk-k^2}{2} \qquad\qquad\qquad\qquad\qquad $$
\begin{equation} R=\frac{mk-k^2}{2}\quad\implies k=\frac{m\pm\sqrt{m^2-8R}}{2}\\ \text{for}\quad \big\lceil\sqrt{8R}\big\rceil\le m \le (2R+1) \end{equation} The lower limit insures that $k\in\mathbb{N}$. The upper limit ensures that $m> k$. $$R=1\implies \lceil\sqrt{8}\rceil=3\le m \le (2+1)=3 \qquad\land\qquad m\in\{ 3\}\implies k\in\{ 2,1\}$$ $$F(3,2)=(5,12,13)\quad\land\quad \frac{30}{30}=1\qquad\qquad\qquad F(3,1)=(8,6,10)\quad\land\quad \frac{24}{24}=1$$
It appears that $R=1$ for perfect triangles but any ratio can be found this way. for example, $R=1.5$ yields \begin{align*} R=1.5 &\rightarrow \space (7,24,25)\quad\space\space (15,8,17)\\ R=2.0 &\rightarrow \space (9,40,41)\quad\space (12,16,20)\quad (24,10,26)\\ R=2.5 &\rightarrow (11,60,61)\quad (35,12,37)\\ R=3.0 &\rightarrow (13,84,85)\quad (16,30,34)\quad (21,20,29)\quad (48,14,50) \end{align*}