6. If the three sides of a right-angled triangle are positive integers, and the numerical value of its perimeter is equal to the numerical value of its area, then it is called a "standard right-angled triangle". Then, the number of standard right-angled triangles is ( ).
(A) 0
(B) 1
(C) 2
(D) infinitely many
6. C.

Let the three sides of the right triangle $\triangle ABC$ be $x, y, z$ $(0 < x \leqslant y < z)$. Then,
$$
x + y + z = \frac{xy}{2}, \quad x^2 + y^2 = z^2.
$$

Therefore, $x^2 + y^2 = \left(\frac{xy}{2} - x - y\right)^2$.
This simplifies to $\frac{xy}{4} + 2 = x + y$, or $(x-4)(y-4) = 8$.
Thus, $x-4=1, y-4=8$
or $x-4=2, y-4=4$.
Hence, $(x, y, z) = (5, 12, 13)$ or $(6, 8, 10)$. These are the two right triangles.