Find all triangles of which perimeter and area are numerically equal

Area = $rs$, where $r=\text{inradius}$ and $s=\text{perimeter}/2$

You can see that $rs=p \implies r=2$

There are infinite triangles with inradius as $2$

The area is $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s$ is the semiperimeter. Thus we get $$(a+b+c)^2 = \frac{a+b+c}{2}(\frac{a+b+c}{2} - a)(\frac{a+b+c}{2} - b)(\frac{a+b+c}{2} - c).$$ We can further simplify this to $$16(a+b+c) = (-a+b+c)(a-b+c)(a+b-c).$$ Let $u = -a+b+c$, $v = a-b+c$, $w = a+b-c$. Then $$16(u+v+w) = u v w.$$ In particular any $u,v$ such that $uv > 16$ give a solution for $w$: $$w = \frac{16(u+v)}{uv-16}.$$ Now for such $u,v,w$ we have that $a = \frac{v+w}{2}$, $b = \frac{w+u}{2}$ and $c = \frac{u+v}{2}$ are the sides of a triangle whose area is equal to its perimeter.

There is some information at Wikipedia. As another answer notes, these are precisely the triangles with inradius 2. But more information is given, for example, that there are exactly five such triangles with integer sides.