Tank Head Volume Calculator: Formula and Examples

A tank head volume calculator estimates the capacity of a formed end, such as an ellipsoidal, hemispherical, or torispherical head. Enter the head dimensions and choose its shape; the calculator returns the head’s volume, which you can add to the cylindrical shell volume to estimate total tank capacity.

Formula

This calculator uses the standard volume formula for a half-ellipsoid head: V = πD²H / 6. Here, V is the head volume in cubic metres, D is the inside diameter in metres, and H is the inside depth from the head’s tangent line to its crown, in metres. The second output converts cubic metres to litres: litres = V × 1,000.

The formula treats the head as half of a complete ellipsoid whose two equal horizontal axes each measure D and whose vertical axis measures 2H. It is appropriate for a 2:1 ellipsoidal head when dimensions are measured at the inside surface. It is not a universal formula for every formed tank end.

Worked Example

Using the calculator’s default dimensions, D = 2 m and H = 0.5 m:

1. Square the diameter: D² = 2² = 4 m².
2. Multiply by the depth and π, then divide by 6: V = π × 4 × 0.5 / 6 = π/3 ≈ 1.047 m³.
3. Convert to litres: 1.047 × 1,000 ≈ 1,047 L.

Rounded as displayed, the head volume is 1.047 m³, or 1,047 L. This is a geometric estimate, not a certified vessel capacity.

What Tank Head Volume Means

A tank head is the curved end closure attached to a cylindrical shell. Its shape affects both capacity and structural behavior. Two vessels with the same diameter can have different head volumes if their head profiles differ. A head calculator therefore needs the correct geometry, not just the tank diameter.

Common head types include:

• Hemispherical: a half-sphere, relatively deep for its diameter. Its volume is πD³/12.
• Ellipsoidal: an elliptical dome. A 2:1 ellipsoidal head has an inside depth of approximately D/4 and volume πD³/24.
• Torispherical: a dished profile made from a crown and a knuckle transition. Its volume depends on the profile dimensions and cannot be determined accurately from diameter alone.
• Flat: a flat end has no dome volume beyond the cylinder’s defined length, though thickness and internal fittings can affect usable capacity.

“Head volume” may mean the internal volume of one end, the combined volume of both ends, or an end’s contribution to a complete vessel. Confirm which quantity is needed before using a result.

When to Use the Estimate

This calculation is useful for preliminary capacity estimates, comparing vessel geometries, and checking whether a stated head volume is plausible. For a vessel with two identical heads, double the calculated result. To estimate the entire tank, add the head volume or volumes to the volume of the cylindrical shell.

For a cylindrical shell with inside diameter D and straight length L, the shell volume is V_shell = πD²L/4. If the vessel has two identical 2:1 ellipsoidal heads, the idealized total is:

V_total = πD²L/4 + 2(πD³/24)

Use consistent units throughout. With metres, the answer is cubic metres; multiply by 1,000 for litres. With feet, the same geometry produces cubic feet; multiply by 7.48052 for US gallons. A US gallon is not the same as an Imperial gallon, so identify which gallon definition is required.

Choosing the Right Dimensions and Shape

Use inside dimensions when estimating the space available for contents. Outside diameter or external head depth will generally overstate internal capacity because the wall has thickness. For a manufactured vessel, drawings or a dimensioned specification are better sources than measurements taken from a rounded exterior surface.

• Measure diameter across the inside of the shell, not around its circumference.
• Measure head depth axially, from the tangent line where the head meets the straight shell to the inside crown.
• Identify the actual profile. Do not assume every dished head is hemispherical or 2:1 ellipsoidal.
• For a partial fill, distinguish total geometric capacity from the volume below a specified liquid level. A partially filled curved head requires a separate level-to-volume calculation.

If the head is torispherical or has a specified crown radius, knuckle radius, straight flange, or unusual profile, use a formula or calculator designed for those details. A simple ellipsoid estimate may be materially different.

Limits, Accuracy, and Common Mistakes

The calculator models an ideal half-ellipsoid. Real heads can depart from that ideal because of their forming profile, thickness, weld transitions, internal coatings, nozzles, or manufacturing tolerances. The result is therefore an approximate geometric capacity, not a substitute for a vessel drawing, calibration chart, or measured fill-volume table when precision matters.

Common errors include entering the radius where the formula expects diameter, mixing inches with metres, using outside dimensions, or treating the depth as the full head height including a straight flange. A radius entered as D makes the calculated result one quarter of the correct value because diameter is squared. Convert all dimensions to one unit before calculation: for example, inches to metres by multiplying by 0.0254, or feet to metres by multiplying by 0.3048.

Also distinguish gross capacity from usable capacity. Working limits, required headspace, process conditions, and vessel orientation affect how much material can safely or practically be held; this geometric calculation does not determine those limits.

Frequently Asked Questions

How do I calculate the volume of a tank head?
Identify the head shape and use its matching formula. For a 2:1 ellipsoidal head, measure inside diameter D and inside depth H, then calculate πD²H/6. The calculator above assumes this half-ellipsoid model.

How do I calculate the volume of two tank heads?
Calculate one head’s volume and multiply by two if both heads are identical. If their dimensions or profiles differ, calculate each separately and add the results.

How do I get the total capacity of a tank?
Add the volume of the cylindrical shell to the internal volume of each head. Use inside diameter and shell length measured between the head tangent lines, and keep all measurements in consistent units.

Is a tank head the same as a tank end cap?
“End cap” is used broadly for an end closure, while “head” commonly refers to a formed vessel end. The terms alone do not specify the profile; determine whether the end is elliptical, hemispherical, torispherical, flat, or another shape.

Can I use this for a torispherical head?
Not reliably from diameter and depth alone. A torispherical head has a crown and knuckle profile, so accurate calculation requires the relevant geometry or a method specifically made for that head type.

Why is the calculated volume different from a tank chart?
A chart may reflect the actual formed profile, vessel dimensions, calibration, and liquid level, while this calculator assumes an ideal ellipsoid. It may also report usable rather than gross volume, or use different measurement units.

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in d | Head diameter | m | 2 | 0
in h | Head depth | m | 0.5 | 0
out volume | Head volume | m^3 | pi * d^2 * h / 6 | 3
out liters | Head volume | L | volume * 1000 | 0