I have found the following equation in my Classical Physics textbook, in the first chapter about Dimensional Analysis: $A=fab$, where $A$ is the surface area of a generic geometric shape, $a,b$ are "characteristic length" of the shape and $f$ is the shape "form factor". It is stated that for every "geometric shape" (I assume it referes to every 2-dimensional shape) there exist such characteristc lengths and $f$. I convinced myself that if $f,a,b$ are found then the equation holds as every other similar shape would just be a scalaed version, which is a linear transformation. However I am not able to convince myself that there exists such characteristic lengths for every shape.
For a square $a,b$ would be the length of the side and thus the equation would be $A=1ll$, for a triangle it would be $A=\frac{1}{2}bh$. I also am not able to figure how such length are choosen in the general case (I imagine the area given by an intergral of a random function?).