Arc length equals that fraction of the full circumference: $\text{arc length} = \dfrac{\text{central angle}}{360}\cdot\text{circumference}$. Substitute the values you know.

Once you have an equation like $6\pi = \dfrac{1}{8}C$, solve for $C$ by undoing the multiplication by $\tfrac{1}{8}$. What do you multiply both sides by?

In Desmos, type 45/360 and see the decimal or simplified fraction; this tells you what fraction of the full circle the given central angle represents.

Once you know the fraction (from step 1), set up the relationship that the given arc length $6\pi$ equals that fraction of the circumference. In Desmos, multiply $6\pi$ by the reciprocal of that fraction (for example, if the fraction is 1/8, type 6*pi*8) and use the output as the circle's circumference.

A full circle measures $360^{\circ}$. The given central angle is $45^{\circ}$.

Compute the fraction of the circle that this angle represents:

So the $45^{\circ}$ arc is $\tfrac{1}{8}$ of the entire circle.

Arc length is a fraction of the full circumference, based on the central angle:

Here, the arc length is $6\pi$ and the fraction is $\tfrac{1}{8}$, so let $C$ be the circumference and write:

Solve the equation

by multiplying both sides by 8:

So

The circumference of circle $C$ is $48\pi$, which corresponds to choice C.