In more detail
For a rectangular pool, volume is length × width × average depth; for a round one, π × radius² × depth; an oval is approximately length × width × depth × 0.785. With a uniform slope, the average depth is (shallow + deep) / 2. In US units multiply cubic feet by 7.48 for gallons; in metric, each cubic metre is 1,000 litres. A filled pool holds a little less than the shell's volume because the water line sits below the top edge.
Why it matters
Chemical doses are given per 10,000 gallons or per cubic metre, so a wrong volume means too little sanitiser or too much. Pump and filter sizing depends on how quickly the whole volume can be circulated, and heaters, fill-up costs and covers are all sized from the same figure.
Example
Rectangle 30 × 15 ft, 3.5 ft to 8 ft deep: average (3.5 + 8) / 2 = 5.75 ft
30 × 15 × 5.75 = 2,587.5 ft³ × 7.48 = 19,356 US gal
Round, 24 ft across, 4 ft deep: π × 12² × 4 = 1,809.6 ft³ = 13,536 gal
Metric, 8 × 4 m, 1.5 m average depth: 48 m³ = 48,000 L
Turnover of 7,480 gal at 60 gal/min: 7,480 / 60 = 124.7 min