How do we know that pi meaningfully exists?

I mean: If we define pi as the ratio between a circle's circumference and diameter, how do we know that the comparison is even possible? Why should the circumference of a circle, a curved bit - even be a multiple of a straight line?

Yes, we can physically take a piece of rope and use it to 'measure' a straight segment, then take the same rope and bend it around a circle and 'measure' the diameter, but how do we know we aren't somehow tricking ourselves in doing so?