By measuring it

In math, it’s a theorem based on certain assumptions and definitions about the distances between points, and what length means. You start with human-made assumptions and follow them wherever they lead. They might or might not describe physical reality though.

The assumptions are pretty well justified based on local observations of the everyday real world that we see. Are they true on a bigger scale, say at astronomical distances? People began to wonder this in the 1800’s, in the era of Gauss and Riemann. There’s another theorem that the interior angles of a triangle add up to 180 degrees, and Gauss (an astronomer as well as a math whiz) actually proposed testing that on astronomical observations. I don’t know if he tried any experiments though. A deviation from 180 degrees would mean that space was curved.

Lo and behold, it turns out that space actually is curved, in the presence of gravitational fields. That was figured out by none other than Einstein, who became world famous when Eddington did an observation during a solar eclipse in 1919 and saw the apparent motion of distant stars when they got lined up with the edge of the sun, confirming Einstein’s prediction. The eclipse was needed for the observation since otherwise the sun would have drowned out the distant stars. But, it was quite a sensitive experiment, maybe not possible in the era of Gauss.

Anyway, the “big” answer to your question is that the ratio being constant is in the end an empirically observed fact, but that on a cosmic scale is mostly only a close approximation, and (even Einstein didn’t foresee this) falls completely apart near very extreme regions like black holes.

Einstein’s theory (“general relativity”) was still an incredible work of genius. As the saying goes, they didn’t call him Einstein for nothing!

Shapes don’t actually exist. They are abstract concepts that we use to describe and make predictions about the material world. We know that the ratio between the circumference and diameter of a circle are always the same because that’s part of how we define a circle.

Because circle all have the same proportions. You can take any circle, and just evenly make it bigger or smaller to make it perfectly overlap with any other circle.

The ratios of shapes only ever change if their proportions change. That’s why every single square also always has the same ratio between it’s side and diagonal (√2).

And the ratio of a rectangles side to it’s diagonal will always be the same, regardless of size, as long as the aspect ratio is the same.

That’s literally what a ratio of a shape lengths are supposed to measure: if you scale two shapes of equal ratios to the same size, they will always be identical, because that’s what ratios are defined to tell you.

And since any circle is completely indistinguishable from any other circle, except for size, all ratios of a circles size will always be identical

There’s kind of a meek proof built of of work Euclid and Archimedes. There’s a paper that goes over it (I know skimmed) https://arxiv.org/pdf/1303.0904

You can approximate the length of any path (including circles) by adding the lengths of many small line segments that follow that path. Making a line segment bigger by some factor, will increase it’s length by the same factor. Therefore, scaling the circle by any factor, increases it’s circumference by the same factor. Scaling a circle is just scaling it’s radius so: Scaling the radius by some factor, changes the circumference by the same factor. That means the ratio between radius and circumference is always constant.

I hope this is decipherable :D

Because we use it predictively enough we’d hsve figured out if it wasn’t by now

We know the circumference of a circle is pi * Diameter (c = pi * d). The diameter of a circle is 2 * Radius (d = 2 * r). Therefore the circumference of a circle is 2 * pi * Radius (c = 2 * pi * r). The ratio between circumference and diameter is pi, which is a constant and therefore doesn’t change even when radius size changes.

How do we know Pi? We have literally known about it for so long that no one has an historical account of who first conceived of it. The oldest example we have is from Babylon, and even then we don’t think they discovered it - just that they already were aware of it. https://www.britannica.com/science/pi-mathematics

The greeks figured this one out, and I tend to believe them.