2:1 Ellipsoidal Heads volume formula
Total volume, exact partial fill, derivation, worked example and a copy-ready fill table.
V = πr²Lcyl + (2/3)πr²a, a = D/4V(h) = seg(r,h)·Lcyl + ½·πh²(3r−h)/3seg(r,h) = r²·acos((r−h)/r) − (r−h)·√(2rh−h²). All closed-form — spreadsheet-implementable with ACOS and SQRT.Derivation
The head pair forms a spheroid: a sphere compressed along the tank axis to depth a = D/4. Compression is an affine map, which preserves volume ratios — so the pair's partial fill is exactly (a/r) = ½ times the spherical cap. Exact, not approximate; our tests confirm it against numerical integration to <0.2%.
Worked example
Take D = 40 in, Lcyl = 100 in. Total capacity: 616.5 gal. At a stick reading of 16 in (40% of the 40 in maximum depth), the filled volume is 228.7 gal — 37.1% of capacity. Check it live on the calculator.
| Depth | Volume |
|---|---|
| 5% | 1.7% |
| 10% | 4.9% |
| 15% | 9% |
| 20% | 13.8% |
| 25% | 19.1% |
| 30% | 24.8% |
| 35% | 30.8% |
| 40% | 37.1% |
| 45% | 43.5% |
| 50% | 50% |
| 55% | 56.5% |
| 60% | 62.9% |
| 65% | 69.2% |
| 70% | 75.2% |
| 75% | 80.9% |
| 80% | 86.2% |
| 85% | 91% |
| 90% | 95.1% |
| 95% | 98.3% |
Need the full inch-by-inch table for specific dimensions? The dip chart generator prints it as a laminate-ready PDF.
FAQ
What is the formula for a 2:1 ellipsoidal heads volume?
Total: V = πr²Lcyl + (2/3)πr²a, a = D/4. Partial fill at depth h: V(h) = seg(r,h)·Lcyl + ½·πh²(3r−h)/3. Derivation and a worked example are above.
Is the fill-ratio table exact?
Yes — computed by the same engine as our calculators (a vessel with shell length = 2.5×D), verified by the automated test suite described on the methodology page.
Can I use this in a spreadsheet?
Yes — the formulas are closed-form (acos and sqrt only). Copy the 5%-step table for quick interpolation, or implement the formula directly for exact values.