Why is Wavefunction Normalization on a Ring Done Using dPhi?

[M-Speezy]

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.

 

[mfb]

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.