Partially filled horizontal cylinder volume
Turn liquid depth into volume with circular-segment geometry and learn why depth percentage is not the same as volume percentage.
What the calculation represents
In a horizontal cylinder, the filled cross-section is a circular segment. Percentage of liquid depth therefore differs from percentage of volume except at empty, half-full, and full. The formula uses inside radius r, straight cylindrical length L, and vertical liquid depth h, where 0 ≤ h ≤ 2r.
- Reference formula
V = L × [r² × arccos((r − h) ÷ r) − (r − h) × √(2rh − h²)]- Unit reference
- 1 L = 1,000 cm³ · 1 m³ = 1,000 L · 1 US gal = 231 in³ = 3.785411784 L
A practical workflow
- Confirm that the main body is a horizontal circular cylinder and separate any dished heads. Measure inside diameter D and straight inside length L.
- With the tank level, measure vertical liquid depth h from the lowest inside point to the surface. It must satisfy 0 ≤ h ≤ D.
- Calculate r = D ÷ 2 and the circular-segment area. The inverse cosine function must use radians.
- Multiply area by L. For meter inputs, multiply cubic meters by 1,000 for liters; calculate heads and internal displacement separately.
Worked example
A tank with a 1.20 m inside diameter and a 2.50 m straight cylindrical body holds about 553 liters at a 0.30 m liquid depth. That depth is 25% of the diameter, but the volume is only 19.6% of the full cylinder’s 2,827 liters.
Liters in a partly filled horizontal cylinder
Calculate the contents from the tank length, inside diameter, and vertical liquid depth.
Reading the answer correctly
Dished or elliptical heads, tilt, baffles, and internal pipework are outside this ideal model. Certified strapping or calibration tables take priority for trade measurement, fuel, hazardous contents, and regulated work.
The depth-volume relationship is S-shaped. Near the bottom, a rise adds relatively little volume; around the center it adds much more per unit of depth; near the top it narrows again. Linear interpolation is only exact at isolated points.
Circular-segment equation
The liquid cross-section is bounded by the surface chord and the lower arc. Multiplying that area by straight cylinder length gives the body volume.
The same expression works above half-full up to h = 2r. Avoid a formula limited to shallow segments and verify that inverse cosine is evaluated in radians.
r = D ÷ 2A = r² × arccos((r − h) ÷ r) − (r − h) × √(2rh − h²)V = A × Lliters = V(m³) × 1,000
Depth and volume do not rise together
Reference values for D = 1.20 m and L = 2.50 m with no dished heads. Rounding depth before calculation can shift the result by many liters.
Swipe horizontally to see every column
| Liquid depth h | % of diameter | % of volume | Liters |
|---|---|---|---|
| 0.00 m | 0% | 0.0% | 0 L |
| 0.15 m | 12.5% | 7.2% | 204 L |
| 0.30 m | 25% | 19.6% | 553 L |
| 0.60 m | 50% | 50.0% | 1,414 L |
| 0.90 m | 75% | 80.4% | 2,275 L |
| 1.20 m | 100% | 100.0% | 2,827 L |
Checks before relying on the number
- Diameter, length, and depth are inside measurements in one unit.
- Depth is vertical, not measured along the curved wall or from a raised access opening.
- The result returns zero at h = 0, 50% at h = r, and full-cylinder volume at h = 2r.
- Tilt, heads, baffles, and fittings are documented; regulated measurement uses the valid tank calibration table.
Practical questions
Does 25% liquid depth mean 25% of the liters?
No. In an ideal horizontal cylinder, 25% of diameter contains about 19.6% of volume, while 75% depth contains about 80.4%.
Do dished heads count as part of cylinder length?
Not as straight cylinder. Use only straight shell length and add the heads from their proper geometry or, preferably, the manufacturer’s calibration table.
What if the tank is tilted?
The liquid surface remains level while depth changes along the tank, so one segment no longer represents the full body. Level the tank or use a tank-specific calibration.
Can this determine fuel quantity for sale?
It is not a trade measurement. Custody transfer and regulated inventory require the applicable calibration, corrections, and procedures for the specific tank and product.
Sources and references
These references support the units, method, or limitations explained in this guide.
- NIST Circular 416: gallonage tables for horizontal cylindrical tanks
Public metrology reference containing tank tables and horizontal-cylinder geometry.
- ISO 12917-1: calibration of horizontal cylindrical tanks
Official standard record for manual calibration methods used with horizontal tanks.
- NIST SI units: volume
Reference for expressing and converting SI volume.
Keep D, L, h, straight-body volume, and head corrections as separate records. That makes a level table traceable and easier to revise.
See how formulas, examples, and updates are reviewed.