Choice C is correct. The volume, $V$ , of a right circular cylinder is given by the formula $V = π r^{2} h$ , where $r$ is the radius of the base of the cylinder and $h$ is the height of the cylinder. It's given that a right circular cylinder has a height of $6$ centimeters. Therefore, $h equals 6$ . It's also given that the cylinder has a base diameter of $22$ centimeters. The radius of a circle is half the diameter of the circle. Since the base of a right circular cylinder is a circle, it follows that the radius of the base of the right circular cylinder is $StartFraction 22 Over 2 EndFraction$ , or $11$ , centimeters. Therefore, $r equals 11$ . Substituting $11$ for $r$ and $6$ for $h$ in the formula $V = π r^{2} h$ yields $upper V equals pi left parenthesis 11 right parenthesis squared left parenthesis 6 right parenthesis$ , which is equivalent to $upper V equals pi left parenthesis 121 right parenthesis left parenthesis 6 right parenthesis$ , or $upper V equals 726 pi$ . Therefore, the volume, in cubic centimeters, of the cylinder is $726 pi$ .

Choice A is incorrect. This is the volume of a right circular cylinder that has a base diameter of $2 StartRoot 22 EndRoot$ , not $22$ , centimeters and a height of $6$ centimeters.

Choice B is incorrect. This is the volume of a right circular cylinder that has a base diameter of $4 StartRoot 11 EndRoot$ , not $22$ , centimeters and a height of $6$ centimeters.

Choice D is incorrect. This is the volume of a right circular cylinder that has a base diameter of $44$ , not $22$ , centimeters and a height of $6$ centimeters.