Inputs
Tank shape, the corresponding inside dimensions – diameter and length or height, and for a rectangular tank width, length and height – plus the measured fill height.
Fluid & thermodynamics · Tanks
Tank contents from the fill height
Work out how much is in a tank from the measured fill height. On a horizontal cylinder that is not proportional to the height but follows the area of a circular segment – the calculator solves it in closed form and additionally shows how many litres one centimetre of level means at the current point.
Plain cylindrical and rectangular geometry. Dished and torispherical heads, sumps, internals, wall thickness and the thermal expansion of the contents are not included; for custody transfer the vessel's calibrated capacity table applies.
VA(h) · L
02
Contents
Set the tank shape and the fill height.
Inputs and method
Tank contents
Work out from a measured fill height how much liquid is in a horizontal or vertical cylinder or a rectangular tank.
Calculation
For a vertical cylinder and a rectangular tank the contents are the base area times the fill height and therefore proportional to it. For a horizontal cylinder the filled cross-section is a circular segment instead: A(h) = r²·arccos((r−h)/r) − (r−h)·√(2rh−h²), multiplied by the length. From the width of the free surface follows additionally how many litres one centimetre of level means at the current point.
Example
A horizontal cylinder of 1,200 mm inside diameter and 1,800 mm length holds 2,035.8 l. At a fill height of 600 mm – half the height – it is exactly 50 % by symmetry, whereas at 300 mm it is only 19.6 %.
Sources and limits: Closed-form circular-segment and cuboid geometry. Dished and torispherical heads, sumps, internals, wall thickness and thermal expansion of the contents are excluded; for custody transfer the vessel's calibrated capacity table applies.
Tank contents in detail
Work out from a measured fill height how much liquid is in a horizontal or vertical cylinder or a rectangular tank.
Why are tank contents not always proportional to the height?
The contents of a tank follow from the filled cross-sectional area times its length. Whether the contents are proportional to the fill height depends solely on whether that cross-sectional area stays constant with height. In a vertical cylinder or a cuboid it does; in a horizontal cylinder the filled area is a circular segment whose width changes with height – and that makes the relation between level and quantity non-linear.
Formula and variables
V = A(h) · L
Horizontal cylinder: A(h) = r² · arccos((r−h)/r) − (r−h) · √(2rh−h²)Vertical cylinder: V = π · r² · hRectangular tank: V = B · L · hLitres per cm = surface width · length · 1 cm
| Symbol / input | Meaning |
|---|---|
| r, D | Inside radius and inside diameter of the cylindrical tank. |
| h | Measured fill height above the lowest point of the interior. |
| L, B | Inside length and, for a rectangular tank, inside width. |
| A(h), V | Filled cross-sectional area and the contents that follow from it. |
Choose the inputs correctly
Tank shape, the corresponding inside dimensions – diameter and length or height, and for a rectangular tank width, length and height – plus the measured fill height.
How to use the calculator
Choose the tank shape – the form then asks for exactly the dimensions that shape needs. Enter the clear inside dimensions, not the outside ones, and then the fill height read from the dipstick above the lowest point of the interior. The calculator returns the contents, fill level and remaining capacity, plus the value that helps most on a horizontal cylinder: how many litres one centimetre of level means at that point.
Worked example
A horizontal cylinder of 1,200 mm inside diameter and 1,800 mm length holds 2,035.8 l. At a fill height of 600 mm – half the height – it is exactly 50 % by symmetry, whereas at 300 mm it is only 19.6 %.
Which value is meant for what?
Contents and remaining capacity are the numbers for planning and reordering. The fill level in per cent is the number that surprises people on a horizontal cylinder and guards against misjudgement – there it differs considerably from the percentage of the fill height. The litres per centimetre, finally, say how sensitive the result is to measurement accuracy.
All dimensions in millimetres, results in litres and cubic metres. 1 m³ equals 1,000 litres; 1 litre equals 1 dm³ or 1,000,000 mm³.
Typical applications
Reading a dipstick on heating-oil, water and process tanks, determining a refill quantity, estimating the residual contents before draining, checking a level gauge, and rough sizing of catch basins and feed vessels.
Assumptions, limits and common mistakes
Closed-form circular-segment and cuboid geometry. Dished and torispherical heads, sumps, internals, wall thickness and thermal expansion of the contents are excluded; for custody transfer the vessel's calibrated capacity table applies.
Common mistake: Do not use simple proportion on a horizontal cylinder: half the height is indeed half the contents, but a quarter of the height is only about a fifth. Do not enter the outside diameter if the wall is thick. And on tanks with dished heads the actual contents exceed the value calculated here, because only the cylindrical part is covered.
Frequently asked questions
Why is half the fill height exactly half the contents, but a quarter is not a quarter?
Because a horizontal cylinder is symmetric about its horizontal mid-plane: what is missing above corresponds exactly to what is added below, so half the height is exactly half. Below that, however, the cross-section narrows towards the bottom, so the lowest centimetres hold much less than the middle ones – at a quarter of the height it is only about 19.6 %.
What is the litres-per-centimetre value for?
It states how much the contents change when the level rises or falls by one centimetre. On a horizontal cylinder it is largest at the middle and tends to zero towards the bottom and the top. That shows how accurate the height measurement has to be: near empty or full a reading error of a few millimetres hardly matters, in the middle it matters considerably.
Does this apply to tanks with dished heads?
Only to the cylindrical part. Dished and torispherical heads add extra volume at both ends that is not captured here, so the actual contents are above the calculated value. For vessels with pronounced heads the manufacturer's capacity table governs.
Can I use this for custody transfer?
No. For commercial transactions the calibrated capacity table of the specific vessel applies, which accounts for manufacturing tolerances, heads, internals and the installation. This calculator gives a geometrically exact but idealised quantity.
Why does a vertical cylinder behave so differently from a horizontal one?
Because in a vertical cylinder the cross-sectional area is constant over the whole height – the contents rise linearly with the level. In a horizontal cylinder the width of the liquid surface changes with height, and it is exactly that change which produces the S-shaped curve.