There’s kind of a 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 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

I think c/nostupidquestions is actually for questions that are kinda stupid. This is too smart of a question. /s

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

There’s always a unit that makes the radius 1, 1cm, 1 foot, 1 furlong, one light year, 1 of something else. So the math for a circle of radius 1 would still hold.

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.

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