Geometry: Circumference of circle without $\pi$

Question

I just found out a way to find out the circumference of the circle without using $\pi$:

$$4\sqrt{(1.8626\cdot r)^2 - r^2}$$ It can calculate up to $2$ decimals equal to the answer got by
using $\pi$.

Please let me know if it works.

Thanks.

Answer 1

Your approximation works because $$2\sqrt{1.8626^2 -1} \approx \pi$$

But the approximation is not exact because otherwise, it would imply $\pi$ is algebraic, but it is not. You can build infinitely many such approximations by replacing $$1.862$$ by any number as close as you want to $$\sqrt{\frac{\pi^2}{4}-1}$$

Answer 2

If we define $x=\frac{\pi}{2}$, $y=x^2-1$ (corresponding to $x=\sqrt{y-1}$) and $z=\sqrt{y}$ use that to rewrite the standard formula for the circumference of a circle we get $$ 2\pi r=2\pi\sqrt{r^2}=4x\sqrt{r^2}=4\sqrt{y-1}\sqrt{r^2}=4\sqrt{r^2(y-1)}=4\sqrt{yr^2-r^2}=4\sqrt{(zr)^2-r^2} $$ which looks like your formula. All that's missing is replacing $z$ with $1.8626$ and only claiming it to be an approximation.

As the definition of $x$, $y$ and $z$ is rather arbitrary, it follows that we could have used other definitions and gotten other approximations.