Ancient geometers took the similarity of circles (in the plane) as self-evident. They knew that if you created shapes with ruler and compass following a formula, then by scaling up or down the new shape would be congruent to the first.
Euclid used this to show a relationship between the area of two circles without knowing about $\pi$. He might have taken his ruler and compass and thrown it against the wall when contemplating the 'perimeter' of a circle.
Years later Archimedes took on the challenge. For more, see
Who First Proved that C / D is Constant?
and for the in-depth historical details,
CIRCULAR REASONING: WHO FIRST PROVED THAT C/d IS A CONSTANT?
where, amazingly, you'll find,
Aristotle’s belief that curves and line segments can not be compared
persisted. Most famously, in his 1637 La Géométrie, René Descartes
Geometry should not include lines [curves] that are like strings, in
that they are sometimes straight and sometimes curved, since the
ratios between straight and curved lines are not known, and I believe
cannot be discovered by human minds, and therefore no conclusion based
upon such ratios can be accepted as rigorous and exact.