Formula for sides of a triangle where the Perimeter equals to the Area

Question

I was wondering if there is a formula that could generate the values of the sides of a triangle where his area equals to his perimeter. I only found that if the triangle is equilateral then $$l=\frac{12}{√3}$$ where $l$ is the side of the triangle.

Thanks for support

Peterix

P.S. There is a similar problem here

Answer 1

Let $S$ be the area of a triancle. Let $P$ be the perimeter of a triancle. Let $r$ be the inscribed circle radius of a triangle.

Since $S=\frac12 r P$ we have $r=2.$ It gives us information for certain types of triangles. for exemples. For a right triangle we have $$r=\frac{a+b-c}{2}=2$$ Since $$c^2=a^2+b^2$$ we have $$a+b=2+\frac{ab}{4}$$