How to find area of this curved shape?
[Math] Find the area of the curved shape
geometryintuition
Related Solutions
Contrary to all other answers, I say yes you can find the area $a$ of a known shape (clover leaf) from the length of its perimeter $p$.
Taking a similar model of the clover leaf, measure its area $A$ and perimeter $P$, using a curvimeter and a planimeter. You can also do that from a digital image (photoscan), but I don't know of ready-made tools for that.
Then, for any clover leaf (of the same shape), this proportionality rule holds:
$$\frac aA=\left(\frac pP\right)^2,$$ so that $$a=A\left(\frac pP\right)^2=F_{clover}p^2.$$
For any shape there is a corresponding conversion factor that you can compute once for all.
For instance, with the picture below, you can estimate an area of $19852$ pixels and a perimeter of $750$ pixels (this is an inaccurate measurement).
Then $F_{clover3hearts}\approx0.0353$, and your leaf has an area of $325$ square units.
The name of the shape depends on the exact curvature in the longitudinal direction and what happens at infinity. You have shown a finite segment, so one cannot say for sure.
If the surface has constant negative Gaussian curvature, it is known as a pseudosphere (the sphere itself has constant positive Gaussian curvature). It has an outer limit where the surface turns parallel to its cross-section, similar to the outermost (largest) circle in the illustration but further down the curve.
If the longitudinal curve is a segment of a parabola or hyperbola, then the shape is an example of a quadric surface. The full curve extends to infinity in both directions.
Best Answer
It looks like the left and right boundaries of your region can be represented by equations of the form $x = f(y)$ and $x = g(y)$ for $0 \le y \le 16$. Then your area is $\int_0^{16} (g(y) - f(y))\ dy$. It also looks like $g(y) - f(y)$ is approximately (but not exactly) $4 + X$, so the area is approximately $16 (4 + X)$.