# Which right triangles have area equal to perimeter?

Which right triangles have integer sides such that the area and perimeter are equal?
asked Jul 10, 2014 in Math

+1 vote
(a,b,c)= (5,12,13), (6,8,10), (6,25,29), (7,15,20), (9,10,17)
answered Jul 10, 2014 by Evoker (610 points)

So the triangle side lengths (a, b, c) for the first one are (5, 12, 13), respectively.

The perimeter is then P = a + b + c = 5 + 12 + 13 = 30 and the semiperimeter, call it s, is therefore 15.

Using Heron's formula to find the area, $$A = \sqrt{s \cdot (s-a)(s-b)(s-c)}$$, we get

$$A = \sqrt{15 \cdot (15-5)(15-12)(15-13)}$$

$$A = \sqrt{15 \cdot (10)(3)(2)}$$

$$A = \sqrt{900}$$

$$A = 30$$

Nice. I wonder what is the general approach to finding these?

Maybe if we start with P = A, or $$a + b + c = \sqrt{s \cdot (s-a)(s-b)(s-c)}$$.

But this is one equation with three unknowns