What is the relation between perimeter and triangle area?

If you double (2 times) the perimeter the area will will be 4 times larger.

Therefore the area is proportional to the square of the perimeter

or

the perimeter is proportional to the square root of area.

The relationship as shown above applies only to triangles with similar proportions, that is when you scale up or down any triangle of fixed proportions.

Other than that requirement, there is no relationship between perimeter and area of any shape of triangle except that it can be stated that the area will be maximum when the sides are of equal length (sides = 1/3 of perimeter).