Why is Wavefunction Normalization on a Ring Done Using dPhi?

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SUMMARY

The normalization of a wavefunction for a particle on a ring is performed using the differential element dPhi, rather than r*dPhi, because the wavefunction is expressed in terms of the angular coordinate phi. In this context, r is a constant representing the radius of the ring, which does not affect the normalization process. The integral of the amplitude squared over the ring must equal 1, and the prefactor in this integral is determined by the specific form of the wavefunction and the limits of integration.

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M-Speezy
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When normalizing a Wavefunction for a particle on a ring why is the normalization only done as dPhi? It's a particle on a ring, so shouldn't it be r*dPhi? This is my thinking, but I do not find other solutions doing this, just ignoring the r part. I understand that for a ring it's just a constant but it still seems important.
 
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Is the wavefunction expressed in terms of a linear distance or in terms of the angle phi?

If you integrate the amplitude squared over the ring the result has to be 1 - the prefactor will depend on the integral you perform.
 

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