Vertical Tank, 2:1 Dished Bottom volume formula
Total volume, exact partial fill, derivation, worked example and a copy-ready fill table.
V = (2/3)πr²a + πr²Hcyl, a = D/4h≤a: V = (r/a)²·πh²(3a−h)/3 | h>a: V = (2/3)πr²a + πr²(h−a)seg(r,h) = r²·acos((r−h)/r) − (r−h)·√(2rh−h²). All closed-form — spreadsheet-implementable with ACOS and SQRT.Derivation
The bottom head is half a spheroid standing on its pole. Stretch a sphere of radius a horizontally by r/a: areas scale by (r/a)², so the head's partial fill is exactly (r/a)² times a spherical cap of radius a. Above the tangent line, linear cylinder math takes over.
Worked example
Take D = 48 in, Hcyl = 60 in. Total capacity: 532.7 gal. At a stick reading of 28.8 in (40% of the 72 in maximum depth), the filled volume is 194.3 gal — 36.5% of capacity. Check it live on the calculator.
| Depth | Volume |
|---|---|
| 5% | 1.4% |
| 10% | 5.1% |
| 15% | 10% |
| 20% | 15.3% |
| 25% | 20.6% |
| 30% | 25.9% |
| 35% | 31.2% |
| 40% | 36.5% |
| 45% | 41.8% |
| 50% | 47.1% |
| 55% | 52.4% |
| 60% | 57.6% |
| 65% | 62.9% |
| 70% | 68.2% |
| 75% | 73.5% |
| 80% | 78.8% |
| 85% | 84.1% |
| 90% | 89.4% |
| 95% | 94.7% |
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 vertical tank, 2:1 dished bottom volume?
Total: V = (2/3)πr²a + πr²Hcyl, a = D/4. Partial fill at depth h: h≤a: V = (r/a)²·πh²(3a−h)/3 | h>a: V = (2/3)πr²a + πr²(h−a). Derivation and a worked example are above.
Is the fill-ratio table exact?
Yes — computed by the same engine as our calculators (2:1 head (depth D/4), shell = 1.25×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.