Paraboloid Tank

At liquid depth h, the horizontal radius grows in proportion to the square root of h. If the full tank has radius R and height H, this relationship can be written as r = R√(h/H). Consequently, shallow layers have relatively small radii, while deeper layers spread across increasingly wider sections of the vessel.

The cross-sectional area is πr², while the radius varies as the square root of depth. Squaring that radius removes the square root, giving A(h) = πR²h/H. Thus, area is directly proportional to liquid height. This linear area-depth relationship is the key geometric feature used when calculating volume at different levels.

Integrating the linearly increasing cross-sectional area from the bottom to the top produces the factor one-half. With A(h) = πR²h/H, the accumulated volume at height H becomes V = ½πR²H. The result is therefore not obtained by treating the tank as a cylinder with constant area, because the cross-section changes continuously with height.

For liquid depth h in a tank with total radius R and height H, integrating the changing areas up to h gives V(h) = ½πR²h²/H. This expression shows that volume depends on the square of liquid depth. It can therefore estimate storage at an intermediate level rather than only the capacity of a completely filled tank.

The essential measurements are the tank’s full interior radius R and vertical height H. Substituting them into V = ½πR²H gives the full storage capacity. For a liquid level below the top, the measured depth h is also required, allowing the partial-volume expression to determine how much liquid the tank contains.

The volume-depth relationship connects a measured liquid level with stored amount, so it supports both filling and draining analysis. Since V(h) is proportional to h², equal changes in volume do not generally produce equal changes in height. Using the formula at successive depths helps determine capacity changes and interpret liquid-level measurements mathematically.