So I am given a circular area 10 square units, and I am given a length, 6 units which the total perimeters of all the shapes must add up to. All shapes are counted separately, so if I have 2 squares of side length l touching each other, their perimeter would still be 8*l* instead of being counted as 6*l* or 7*l*.
Shapes can be of any shape and any size as long as they have no reflex angles. I need to know the minimum number of shapes I would need to fill up the area completely (or almost completely). How to find a general solution, for any area A and length L would also be appreciated.
I think something is missing from the question, because as stated it's trivial and weird.
It's clear that you should only use one shape, if you have two shapes which overlap (maybe just their borders), you can replace them with one with one, the interior of which is the union of the interior of the two hape (and any common borders).
But your example figures makes for an impossible construct. The optimal raio between area and circumference is achieved by a circle, and a circle with a circumference of $6$ has a radius of $\frac{3}{\pi}$ and an area of $\frac{9}{\pi}\approx 2.86478897565$ far from the $10$ you were supposed to cover.