For any two segments of $2$ different circles, how do we know that the angle made (we call it one radian for all circles) by arc length (equal to length of radius) to radius would be same as that of another circle's.

Also, we say that pi is ratio of circumference to $2$ times the radius. How do we confirm that or prove that it's true for all circles?

  • 2
    $\begingroup$ All circles are scaled versions of the unit circle. Scaling uniformly in all directions preserves angles. Visually, overlay one circle over a smaller one. Draw some triangles, make some angles,.... they have to be the same $\endgroup$
    – FShrike
    Commented Aug 11, 2021 at 10:16
  • $\begingroup$ This question might interest you. $\endgroup$
    – drhab
    Commented Aug 11, 2021 at 10:17
  • $\begingroup$ The what about one radian? How is the angle same for all the circles when arc length is equal to radius? We know it visually, but how can we prove it Mathematically? $\endgroup$ Commented Aug 11, 2021 at 15:34


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