Constraint Equation of a rolling of cylinder

[Molar]

I was going through this example from by book, where a cylinder of radius 'r' is rolling inside of another cylinder of radius 'R' without slipping.It says the constraint equation should be
rθ = (R-r)φ
where θ = angle of rotation of cylinder of radius 'r'
φ = angle subtended at the centre of the cylinder of radius 'R'

I am not getting this. Shouldn't it be rθ = Rφ ...? To me it looks like if we deduct 'r' from 'R' then
rθ will be greater than (R-r)φ . Please correct me.
Thanks.

 

[PeroK]

I was going through this example from by book, where a cylinder of radius 'r' is rolling inside of another cylinder of radius 'R' without slipping.It says the constraint equation should be
rθ = (R-r)φ
where θ = angle of rotation of cylinder of radius 'r'
φ = angle subtended at the centre of the cylinder of radius 'R'

I am not getting this. Shouldn't it be rθ = Rφ ...? To me it looks like if we deduct 'r' from 'R' then
rθ will be greater than (R-r)φ . Please correct me.
Thanks.

Try an example where $R = 4$ and $r = 1$

How many times does the smaller cylinder rotate to circumscribe the inside of the larger cylinder once?