How many triangles with whole number leg lengths are there such that area and the perimeter is equal? I've tried to use Heron's formula to approach the problem , but it doesn't make any sense .I also tried to guess the lengths and I got two triangles , one of them is (5,12,13) and the second is (6,8,10).
So,I hope you can help me to find out  "Is there efficient way to solve this problem ?"

Trying with Pythagorean triplets with sides $ (2mn, m^2-n^2,m^2+n^2),$ 
your condition leads to 
$$ 2 m^2 + 2 mn = mn ( m^2-n^2) ; \quad n (m-n) = 2; $$
or
$$ n = (m + \sqrt{ m^2-8})/2 $$
which gives an infinite set including
$$ (m,n) = (3,2), \sqrt2 (2,1),..  $$
