Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number?
I suppose it would depend one what would be defined as "circular curves", as well as where you would measure the diameter on a shape that is not infinitely radially symmetrical like a circle. So I'd like to keep the question open to different possibilities compared with others.
What brings me to ask to the question in the first place is imagining the irrational or transcendental property emerging from curves.
One scenario to consider would be a square with rounded corners. If it was a ratio of the shape's perimeter to the the diameter measured from one corner to a corner diagonally opposite of it, would it result in an irrational number?
How about the same scenario, but it's the ratio of the shape's perimeter to the diameter measured from the center of one flat side to the opposite flat side?
The reason I have not figured this out for myself is that I don't know how to get the perimeter from a square with curved corners, or where to measure from in the case where the diameter is measured from the corners. (I don't know how to get the exact points on the curve.)
If within the class of curves you want to consider, there exists a 1-parametric family of curves (i.e., you can sometimes continuously deform such a curve without making it “invalid”) and not all curves in thus family have the same perimeter/diameter ratio, then the answer is definitely: No, some such curves have a rational ratio.
Note that this is specifically true for your example class of curves: squares with rounded corners. Indeed, such a shape with straight edge parts of length $a$ and rounded corners if radius $r$ has perimeter $4a+2\pi r$ and diameter $a\sqrt2+2r$ so that the ratio is $$\frac{4a+2\pi r}{a\sqrt2+2r}$$ Or with $t:=\frac ar$, $$\frac{4t+2\pi}{t\sqrt2+2}.$$ For $t=0$, this is $\pi$, and as $t\to\infty$, the ratio approaches $2\sqrt2$. Any value in between, including infinitely many rational values, is obtained for a suitable $t$. One such rational value is $3$. Can you find the $ t$ for that?