Liquid in a Horizontal Cylindrical Tank

Finds how much liquid a cylinder lying on its side holds, from the depth of the liquid. The cross-section is a circular segment, so the volume is not proportional to the depth: it fills fastest around the halfway mark and slowest at the top and bottom.

In an upright tank, half the depth means half the contents. A cylinder lying on its side does not behave that way. Its cross-section is a circle, narrow near the bottom and the top and widest across the middle.

The cross-section below the surface is a circular segment. Its area multiplied by the length of the tank gives the volume held.

\theta = 2\arccos\dfrac{r - h}{r} \qquad V = \dfrac{r^2}{2}(\theta - \sin\theta) \, L

$\theta$ is the central angle up to the surface, in radians
$r$ is the radius of the tank
$h$ is the depth of the liquid
$L$ is the length of the tank
$V$ is the volume held

The term $\dfrac{r^2}{2}\theta$ is the area of the sector. Subtracting $\dfrac{r^2}{2}\sin\theta$ , the triangle formed by the centre and the two ends of the surface, leaves the segment alone.

Example

The default input is a tank 100 cm in diameter and 200 cm long, filled to a depth of 30 cm.

The central angle is $2\arccos(20 \div 50)$ , about 2.3186 radians. The segment area is about 1981.68 cm², and multiplying by the length of 200 gives about 396.3367 L. The full capacity is about 1570.7963 L.

Here is the point worth remembering. The depth is 30% of the diameter, yet the contents are only 25.23% of capacity. Reading the contents off the depth by eye overestimates them.

The surface is $2\sqrt{h(2r - h)}$ wide, about 91.6515 cm.

Symmetry

A depth of 70 cm holds about 1174.4596 L. Added to the 396.3367 L at 30 cm, that comes to exactly the full 1570.7963 L.

This always holds, because the two cases are mirror images of one another. And at half the depth, 50 cm, the tank holds exactly half its capacity.

Points to watch

Dished ends are not included. Real tanks usually bulge outwards rather than ending flat, so they hold slightly more than this. Where a calibrated figure is required, use the strapping table for the individual tank.

The depth cannot exceed the diameter, and the tank is assumed to be level. Any tilt changes how the cross-section is cut.

This is the calculation behind dipping a fuel tank, a road tanker or a water cistern. Since depth and volume are not proportional, the marks on a dipstick are not evenly spaced.