Spherical Tank volume formula
Total volume, exact partial fill, derivation, worked example and a copy-ready fill table.
V = (4/3)πr³V(h) = 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 spherical cap — derivable by integrating circular slices π(2rh−h²)dh from 0 to h. At half depth it gives exactly half the sphere; at 25% depth only ~15.6%, the strongest nonlinearity of any common tank.
Worked example
Take D = 100 in. Total capacity: 2,266.7 gal. At a stick reading of 40 in (40% of the 100 in maximum depth), the filled volume is 797.9 gal — 35.2% of capacity. Check it live on the calculator.
| Depth | Volume |
|---|---|
| 5% | 0.7% |
| 10% | 2.8% |
| 15% | 6.1% |
| 20% | 10.4% |
| 25% | 15.6% |
| 30% | 21.6% |
| 35% | 28.2% |
| 40% | 35.2% |
| 45% | 42.5% |
| 50% | 50% |
| 55% | 57.5% |
| 60% | 64.8% |
| 65% | 71.8% |
| 70% | 78.4% |
| 75% | 84.4% |
| 80% | 89.6% |
| 85% | 93.9% |
| 90% | 97.2% |
| 95% | 99.3% |
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 spherical tank volume?
Total: V = (4/3)πr³. Partial fill at depth h: V(h) = 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 (any sphere (proportion-independent)), 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.