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Choice C is correct. The base of a cylinder is a circle with a diameter equal to the diameter of the cylinder. The volume, $V$, of a cylinder can be found by multiplying the area of the circular base, $A$, by the height of the cylinder, $h$, or $V=Ah$. The area of a circle can be found using the formula $A=\pi r^2$, where $r$ is the radius of the circle. It’s given that the diameter of the cylinder is $8$ inches. Thus, the radius of this circle is $4$ inches. Therefore, the area of the circular base of the cylinder is $A=\pi {\left(4\right)}^2$, or $16\pi$ square inches. It’s given that the height $h$ of the cylinder is $12$ inches. Substituting $16\pi$ for $A$ and $12$ for $h$ in the formula $V=Ah$ gives $V=16\pi \left(12\right)$, or $192\pi$ cubic inches.

Choice A is incorrect. This is the area of the circular base of the cylinder.

Choice B is incorrect and may result from using $8$, instead of $16$, as the value of $r^2$ in the formula for the area of a circle.

Choice D is incorrect and may result from using $8$, instead of $4$, for the radius of the circular base.