When to Apply Circular Convolution Formulas?
Circular convolution can be done using FFTs, which is a O(NLogN) algorithm, instead of the more transparent O(N^2) linear convolution algorithms.  So the application of circular convolution can be a lot faster for some uses.

However, with a tiny amount of post processing, a sufficiently zero-padded circular convolution can produce the same result as a linear convolution, while still being a lot faster when using FFTs.  This is because the tail portion of a sufficiently long zero-padded convolutional result is all zeros, rather than being a non-zero tail result that mixes/sums with the beginning of the convolution result when doing circular convolution, which can make a mess out of non-perfectly-periodic inputs.  

For sequences of windows of data, one can extend this to overlap-add or overlap-save FFT fast linear convolution.