Frustum (Tapered) Tank volume formula
Total volume, exact partial fill, derivation, worked example and a copy-ready fill table.
V = πH(R²+Rr+r²)/3V(h) = πh(ρ²+ρr+r²)/3, ρ = r+(R−r)·h/Hseg(r,h) = r²·acos((r−h)/r) − (r−h)·√(2rh−h²). All closed-form — spreadsheet-implementable with ACOS and SQRT.Derivation
A frustum is a cone with the tip cut off; the standard volume identity applies at any level once you interpolate the wetted-surface radius ρ linearly between the bottom and top radii. Buckets, hoppers and tapered vats are all this one formula.
Worked example
Take Dbtm = 20 in, Dtop = 40 in, H = 50 in. Total capacity: 158.7 gal. At a stick reading of 20 in (40% of the 50 in maximum depth), the filled volume is 39.5 gal — 24.9% of capacity. Check it live on the calculator.
| Depth | Volume |
|---|---|
| 5% | 2.3% |
| 10% | 4.7% |
| 15% | 7.4% |
| 20% | 10.4% |
| 25% | 13.6% |
| 30% | 17.1% |
| 35% | 20.9% |
| 40% | 24.9% |
| 45% | 29.3% |
| 50% | 33.9% |
| 55% | 38.9% |
| 60% | 44.2% |
| 65% | 49.9% |
| 70% | 55.9% |
| 75% | 62.3% |
| 80% | 69% |
| 85% | 76.2% |
| 90% | 83.7% |
| 95% | 91.6% |
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 frustum (tapered) tank volume?
Total: V = πH(R²+Rr+r²)/3. Partial fill at depth h: V(h) = πh(ρ²+ρr+r²)/3, ρ = r+(R−r)·h/H. Derivation and a worked example are above.
Is the fill-ratio table exact?
Yes — computed by the same engine as our calculators (a 2:1 taper (top diameter twice the bottom)), 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.