🔢 Formula Handbook

Tank Volume Formulas by Shape

Q: Why do the formulas give 'geometric volume' instead of 'working capacity'?

Because working capacity depends on how you fill the tank. A propane tank is legally capped at 80% of water volume. A heating oil tank is practically capped at 90% of geometric volume to leave ullage for thermal expansion. A fish tank is filled to within ½ inch of the rim. Geometric volume is the ground truth — you apply the fill rule after. Always know both numbers.

Q: What is the difference between a frustum and a truncated cone?

They are the same thing. Frustum is the engineering term (from Latin for 'piece cut off'), truncated cone is the plain-language description. V = (πH/3)(R² + Rr + r²) works for both. The term 'cone bottom tank' usually means a frustum hopper (parallel top and bottom circles) welded under a cylindrical shell.

Q: Do I really need all these formulas, or is the tank calculator enough?

The calculator is enough for 95% of the work — and it handles the hard cases (partial fill, dished heads, tilt) that hand formulas cannot. Keep this sheet for: (1) verifying the calculator's output against a known formula, (2) quick mental estimates in the field before you reach a computer, (3) building your own spreadsheet or integration. The formulas are the source of truth the calculator runs.

Q: Why is 231 the magic number for gallons?

One US liquid gallon is defined by US federal law as exactly 231 cubic inches — since 1707 in Britain, adopted by the US in 1832. It is not an approximation. So every tank volume in cubic inches, divided by 231, is exact US gallons. Imperial gallons are 277.42 cubic inches, about 20% larger. Metric gallons are not a real unit.

Every tank shape you will encounter — vertical cylinder, horizontal cylinder with partial fill, rectangle, sphere, capsule, oval stadium, cone frustum and dished heads. Each shape gets the formula, variable definitions, three worked examples, and an Excel single-line formula you can copy and paste.

Pick a tank shape to see its formula, variables, examples and Excel version

🪙

Vertical Cylinder

All lengths in same unit — result is cubic that unit

V = π × r² × H

Where

• r = inside radius (r = D/2)
• H = inside height of water/shell

Partial Fill (If Applicable)

Same formula — just change H to fill depth. No segment math needed.

Vertical cylinders are linear: half depth = half volume. This is the simplest partial fill case.

3 Worked Examples

Example 1: Diameter 36 in, height 71 in (275 gal class water tower)

π × 18² × 71 = π × 324 × 71

72,223 in³ ÷ 231 = 312.7 gal (nameplate rounds to 275 because of dished head)

Example 2: Diameter 1.5 m, height 3 m (farm cistern)

π × 0.75² × 3 = π × 0.5625 × 3

5.30 m³ = 5,300 L = 1,399 US gal

Example 3: 55 gallon drum: diameter 22.5 in, height 33.5 in

π × 11.25² × 33.5 = π × 126.56 × 33.5

13,311 in³ ÷ 231 = 57.6 gal (nominal 55 gal, a little headroom left)

Excel / Google Sheets Formula

=PI()*(D/2)^2*H/231

D = diameter in inches, H = fill height in inches → gallons

Dish Head Formulas — The Extra Volume Every Tank Calculator Should Show Separately

Most real tanks are not just a cylinder — they have curved heads that bulge outward and add volume. Here is the contribution of each head type, per head, in terms of the shell diameter D.

Flat Head

V = 0 (no bulge)

Simplest — no extra volume. Tanks with welded flat plates on the ends.

Per head: 0

Hemispherical Head

V = (2/3) × π × r³

ASME pressure vessels, LPG bullets. Two make a full sphere.

Per head: ~0.5236 × D³

2:1 Elliptical Head

V = (1/3) × π × r³

Most common ASME pressure-vessel standard. Depth = D/4.

Per head: ~0.2618 × D³

Torispherical (ASME F&D)

V ≈ 0.0809 × D³

Shallow dished heads on horizontal tanks. Radius ≈ D × 0.8.

Per head: ~0.0809 × D³

💡 Heads add up fast. Two hemispherical heads on a 27" diameter tank add about 14,314 in³ total = 62 gallons. Two flat heads add zero. The homepage tank calculator separates shell volume from head volume so you can see exactly what the curved ends are contributing — and compare head types side by side.

Unit Conversion Quick Reference

Every formula on this page works as long as all inputs are in the same unit. These are the 12 conversions you will reach for most often.

gal → in³

1 US gallon

= 231 cubic inches (exact)

gal → L

1 US gallon

= 3.78541 liters (exact)

gal → ft³

1 US gallon

= 0.13368 cubic feet

ft³ → gal

1 cubic foot

= 7.48052 US gallons

ft³ → L

1 cubic foot

= 28.3168 liters

m³ → gal

1 cubic meter

= 264.172 US gallons

m³ → L

1 cubic meter

= 1,000 liters (exact)

in → cm

1 inch

= 2.54 cm (exact)

gal → lb (water)

1 US gallon of water

= 8.345 lb

gal → fl oz

1 US gallon

= 128 fluid ounces (US)

m³ → ft³

1 cubic meter

= 35.3147 cubic feet

L → ft³

1 liter

= 0.0353 cubic feet

💡 Best Practice: Always Use Inside Dimensions

Every formula here calculates the space that holds liquid — the inside of the tank. Wall thickness eats into that space. Subtract twice the wall thickness from each outside dimension you measure. For steel tanks that is typically 0.125–0.25 inches. For aquarium glass, 6–10 mm. For plastic IBC totes, 3–4 mm.

⚠️ Warning: Don't Forget the Heads or the Fill Rule

A horizontal cylinder calculation without heads misses 14–20% of capacity for ASME pressure vessels. And the geometric volume is not usable capacity — propane tanks are capped at 80%, heating oil at 90%. Nameplate numbers round up. Always compute geometric volume, then subtract both head omission AND the fill rule.

❌ Don't Guess — Use the Calculator for Hard Cases

The horizontal cylinder partial-fill formula requires integral calculus (circular-segment integral). The tilt correction formula requires integrating depth along the whole shell. If your tank is tilted greater than 2° or has dished heads or baffles — skip the hand math, use the full tank calculator on the homepage. It handles the hard cases that the formulas here are too simple to capture.

Frequently Asked Questions

Why do you have a separate shape for 'Capsule' when it is just a cylinder plus two hemispheres?▼

Exactly — that is the capsule formula: πr²L + (4/3)πr³. But remembering that two hemispherical heads equal one full sphere saves you the mistake of calculating them twice. The shape exists because the tank calculator and many steel tank suppliers call this shape 'capsule' or 'bullet tank', and searching for it under 'cylinder with heads' is awkward.

Can these formulas handle tilted tanks?▼

Not directly. Tilting changes which part of the liquid sees which cross-section at each height. For small angles (<2°), the cylindrical formulas still work with a height adjustment. For 5°+ or irregular shapes, you need either the tilt-corrected calculator on the homepage or a manual field strapping. The geometry gets you 90% of the way; tilt is the remaining 10%.

Why do the formulas give 'geometric volume' instead of 'working capacity'?▼

Because working capacity depends on how you fill the tank. A propane tank is legally capped at 80% of water volume. A heating oil tank is practically capped at 90% of geometric volume to leave ullage for thermal expansion. A fish tank is filled to within ½ inch of the rim. Geometric volume is the ground truth — you apply the fill rule after. Always know both numbers.

What is the difference between a frustum and a truncated cone?▼

They are the same thing. Frustum is the engineering term (from Latin for 'piece cut off'), truncated cone is the plain-language description. V = (πH/3)(R² + Rr + r²) works for both. The term 'cone bottom tank' usually means a frustum hopper (parallel top and bottom circles) welded under a cylindrical shell.

Do I really need all these formulas, or is the tank calculator enough?▼

The calculator is enough for 95% of the work — and it handles the hard cases (partial fill, dished heads, tilt) that hand formulas cannot. Keep this sheet for: (1) verifying the calculator's output against a known formula, (2) quick mental estimates in the field before you reach a computer, (3) building your own spreadsheet or integration. The formulas are the source of truth the calculator runs.

Why is 231 the magic number for gallons?▼

One US liquid gallon is defined by US federal law as exactly 231 cubic inches — since 1707 in Britain, adopted by the US in 1832. It is not an approximation. So every tank volume in cubic inches, divided by 231, is exact US gallons. Imperial gallons are 277.42 cubic inches, about 20% larger. Metric gallons are not a real unit.

Every Formula Here Is the Math the Calculator Runs

These formulas are ground truth — the source code of the homepage tank calculator. When the calculator says 268 gallons, that is the π × r² × oval integral + head correction this page teaches you. Compare your hand calculation to the calculator's and you have a quality check on your input. Also visit the calculators hub for specialized tools for each shape — horizontal cylinder, rectangular, sphere/capsule, and more.

Open the Tank Calculator Browse All Guides

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