How many gallons are in an 18x52 inch round pool?

Volume of a cylinder = radius squared * PI * height

Your pools radius is 9ft, and the height is 4ft.

Volume (in sq ft) = 81 * 3.14 * 4 = 1017 cubic feet.
There are 7.5 gallons in a cubic foot, so the total gallons is 1017 * 7.5 = 7640 gallons