# smaller circle into larger circle : find length of common arc

Let a circle of radius $r$ be contained in a larger circle of radius $R$ such that the two circles touch.

What is the length, in radians of the common arc, in blue?

I think the solution is related to the proportion of the area.
But I can't go further.

Edit
When I say the circles touch, I mean that $$R - r = \text{distance between the centers of the two circles}. \tag{1}$$ For a larger inscribed circle, the common arc seems longer

Is this just an illusion?
Does condition (1) force the two circles to be tangent?

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If they just "touch" (if they are tangent), they have a single point in common. Arc length = 0. – Yves Daoust Jun 4 '14 at 19:43
@YvesDaoust I made my question more precised. Does your assertion still hold with my precisions? – Nicolas Essis-Breton Jun 4 '14 at 20:27
If two circles intersect along any arc of positive length, they have to have the same radius. – Alexander Dunlap Jun 4 '14 at 20:28
Maybe you consider the digital circle (formed of discrete pixels), which effectively share several pixels. In the continuous domain, tangent circles have a single point in common. – Yves Daoust Jun 5 '14 at 20:53
Your mentioned “illusion” provides a good example of discontinuity. As r gradually tends to R, the “arc length in common” jumps suddenly from 0 to 2πR. – Mick Jun 12 '14 at 16:02