Do circles exist

Question

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of the graph in the first quadrant , so you have a fourth of the circle. Now the area of this section should be $$\frac{\pi}{4}$$ So we take the definite integral $$\int_0^1{\sqrt{1-x^2}}\,dx$$ to get the area under this section. To do this numerically, we take the anti-derivative which is $$\frac{1}{2}\left(x\sqrt{1-x^2}+\operatorname{arcsin}(x)\right)$$ evaluated from 0 to 1, which gives us $\pi/4$. Now just being curious, the series expansion of $\operatorname{arcsin}(x)$ converges for $|x| < 1$. So once 1 is placed into $\operatorname{arcsin}$, does it only give us a rounded answer which never ends? Does this mean $\pi$ really does never end, which would mean the area of a circle is undefined making circles impossible? Probably flawed but still makes me think.

Answer 1

Circles are clearly defined, as geometrical loci.

What is problematic is their area. It does exist, of course, as the "inside part", but what about its size?

This is a problem that has been around almost since circles were studied (see here).

Its size exist as a real number. Yes, it is problematic. But not enough for it not to exist mathematically.