Capsule Tank (Hemispherical Heads) volume formula
Total volume, exact partial fill, derivation, worked example and a copy-ready fill table.
V = πr²Lcyl + (4/3)πr³V(h) = seg(r,h)·Lcyl + 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 shell contributes a circular segment per unit length; the two hemispherical heads together form a complete sphere, whose partial volume at depth h is the classic spherical-cap formula — so the head term needs no orientation bookkeeping at all.
Worked example
Take D = 37 in, Lcyl = 74 in. Total capacity: 459.3 gal. At a stick reading of 14.8 in (40% of the 37 in maximum depth), the filled volume is 169.1 gal — 36.8% of capacity. Check it live on the calculator.
| Depth | Volume |
|---|---|
| 5% | 1.6% |
| 10% | 4.6% |
| 15% | 8.6% |
| 20% | 13.3% |
| 25% | 18.6% |
| 30% | 24.3% |
| 35% | 30.4% |
| 40% | 36.8% |
| 45% | 43.4% |
| 50% | 50% |
| 55% | 56.6% |
| 60% | 63.2% |
| 65% | 69.6% |
| 70% | 75.7% |
| 75% | 81.4% |
| 80% | 86.7% |
| 85% | 91.4% |
| 90% | 95.4% |
| 95% | 98.4% |
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 capsule tank (hemispherical heads) volume?
Total: V = πr²Lcyl + (4/3)πr³. Partial fill at depth h: V(h) = seg(r,h)·Lcyl + 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 capsule with shell length = 2×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.