Is there a relation for when a circle intersects more than half the perimeter/circumference of another circle?

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Is there some nice formula or algoritm for determining when a circle "hides"/intersects more than half of the perimeter of another circle, in a circle-circle interaction.

Example image: Two example circles on Wolframalpha

A way to either determine if more or less than half of the perimeter is "hidden" or a way to fully calculate the hidden perimeter length is desired.

radii, distance between circles, centers and intersection points (in a x-y coordinate system) are known.

Thanks in advance!

Regards