I think you are operating under the mistaken and oversimplified belief that the perimeter of a shape provides more information than it does in determining the area of the shape.
In theory, you need to know every thing about the curve of the shape-- not just it's length/perimeter but every twist turn and angle.
Of course, if the shape has a general pattern or formula than one can probably determine a lot of this information for relatively little data. For example if you know the shape is a polygon, it's enough to know the lengths of all of the sides and the measurement of all the angles. And if you know all but one of the angles you can determine that final angle and if you know all the angles and all but one of the sides you can determine the final side.
But, in general you need to know a lot more than just the total perimeter and the number of sides.
But then again if you know more information about the shape you can often determine more information. If for example you know it is an equilateral $n$-gon and you know the perimeter is $P$ you know the sides are all $P/n$. (However if $n > 3$ you don't know the angles so you don't know the area.) If you know it is an equiangular $n$-gon you know the the total angle and there for every angle. If you know it is a rectangle then you know all the angles are right angles and it is a parallelagram so all you need for the area is the two side (width and height). If you know the perimeter you only need one side and you can determine the other side.
So, your question.
Perimeter = $P$.
Triangle: With sides $a,b,c: a+b+c = P$ and angles $A,B,C: A+B+C = 180$. Angle $A$ is opposite $a$, $B$ opposite $b$ and $C$ opposite $c$.
We don't have enough information. If we had two sides, we can get the third from the perimeter and from that we can use trig identities to get the angles. Or if we had two angles, we could get the third and then we could get the trig propotional lengths and from the perimeter we can get the sides. Or if we had one side and one angle using trig and the restriction of the perimeter we can the rest.
Basically: If we have $2$ out of the six pieces of information and the perimeter, we can get the remaining four pieces of information, and from the six pieces of information we can get the area.
The six pieces of information are related via:
$\frac {\sin A}a = \frac {\sin B}b = \frac{\sin C}c; A+B+C = 180; a+b+c = P$ and also $a^2 + b^2 + 2ab\cos C = c^2$, $b^2 + c^2 + 2bc\cos A = a^2$, or $a^2 + c^2 + 2ac\cos B$ . (Any two of the variables will get you the remaining four.)
The area of the triangle = $\frac 12 *base*height = \frac 12 *a* ( {\sin C}*b)$ or any of the other combinations of two sides and the angle between them.
Equilateral triangle:
In this case $s= a = b = c = P/3$ and $A = B = C = 60$ so the area is $\frac 12(\frac P3)^2\sin 60 = (\frac P3)^2\frac{\sqrt{3}}4= s^2 \frac{\sqrt{3}}4$.
Quadrilateral:
If you don't know whether it is a rectangle or a parallegram there is no formula in general. But you can know that if the sides are $a,b,c,d$ and the angles are $A,B,C,D$ (capitals opposite lower case) you know: $a + b + c + d = P$ and $A+B+C+D= 360$. And you can calculate the area by "cutting" it into two triangles and figuring the area of those and adding them.
Rectangle:
You know the angles are right angles and that it is a parallelagram so $a =c$ and $b = d$ so $2a + 2b = P$ and $b = \frac {P- 2a}2$ and the area is therefore $a*b = a*\frac{P - 2a}2 = \frac P{2a} - a^2$.
Square:
$s = P/4$ and area is $s^2 = \frac {P^2}{16}$
It's worth noting the area of a regular $n$-gon.
An regualar $n$-gon has $n$ sides all equal so $s = \frac Pn$.
And regular $n$-gon has $n$ angles. These angles add up to $({n-2})*180$ because we can can cut the $n$-gon into $n-2$ triangles whose angles are each $180$. So each angle of the $n$-gon is $\theta = \frac{n-2}n * 180$.
We can slice the $n$-gon into isoceles triangle wedges. Each isoceles tringle slice has a base of length $s = \frac Pn$. The base angle is $\frac {\theta}2 = \frac{n-2}n*90$. So the height of the triangle is $height = opposite = adjacent*\frac {opposite}{adjacent} = (\frac 12 s)*\tan(\frac{\theta} 2)$.
So the area of the triangle is $\frac 12*s* (\frac 12 s)*\tan(\frac{\theta} 2)=\frac 14 s^2 \tan(\frac{\theta} 2)$.
So the area of a regular $n$-gon is $n*\frac 14 s^2 \tan(\frac{\theta} 2)=\frac 1{4n}*P^2\tan(\frac{n-2}n*90)$.
For $n=3$ we get $\frac 1{12}*P^2 \tan 30 = \frac 1{12}*P^2*\frac {\sqrt 3}3=(\frac P3)^2\frac{\sqrt{3}}4 = s^2\frac{\sqrt{n}}4$ which is what we had for equilateral triangle above.
Likewise if $n=4$ we get $\frac 1{16}P^2 \tan 45 = (\frac {P}4)^2=s^2$ as we had for squares.
Believe it or not:
Area of circle = $\lim_{n\rightarrow \infty} \frac 1{4n}*P^2\tan(\frac{n-2}n*90) =P^2 \lim_{n\rightarrow \infty} \frac 1{4n}\tan(\frac{n-2}n*90) =P^2 *\frac 1{4\pi}= (\frac {P}{2\pi}) ^2 \pi = \pi r^2$
Best Answer
The question is where do you start with the definition of "area". As far as I know, there are two options:
Develop a general notion of a measure in the plane, say, in the spirit of the Lebesgue integral. Then the formula for the measure of a rectangle comes as a part of the definition of the measure. The hard part of the theory is to construct a well-defined set-function on a certain algebra of subset of the plane (or $n$-space, there is no fundamental difference between the two constructions).
You can also start with axioms of Euclidean geometry (which do not include anything about areas!) and then develop a concept of area for general (not only convex) polygonal regions in the plane. If your objective is to present something that high school students or typical undergraduate students (in a Euclidean geometry class) can comprehend, then this is a way to go. Take a look at the book
E. Moise, "Elementary geometry from advanced standpoint", especially section 13.5, where he shows that area of any square (with side $a$) is $a^2$, starting merely with the normalization that area of the unit square is 1. Along the way, Moise develops the notion of area of planar polygons from scratch. The most difficult part of his proof (that area exists) is to show that area of a polygon does not depend on triangulation.
To conclude: all you need is the set of (Hilbert's) axioms of Euclidean geometry (since, as Hilbert observed, Euclid's set of axioms is incomplete). You also need a notion of real numbers (which Euclid did not have either), namely, ordered field axioms and the completeness axiom.