The spherical cap
A horizontal bullet's two hemispherical heads make exactly one sphere, so the head contribution uses the same cap formula as a full sphere — the cylinder in between contributes a circular segment times its length.
The most non-linear tank there is
Fill a sphere to a quarter of its diameter and it holds only about 16% of its volume. The curve is brutal at both ends and steep through the middle:
| Level (% of diameter) | Volume (% of capacity) |
|---|---|
| 10% | 2.8% |
| 20% | 10.4% |
| 25% | 15.6% |
| 30% | 21.6% |
| 40% | 35.2% |
| 50% | 50.0% |
| 60% | 64.8% |
| 70% | 78.4% |
| 75% | 84.4% |
| 80% | 89.6% |
| 90% | 97.2% |
Between 40% and 60% of depth the vessel gains roughly 29% of its capacity. Between 0% and 20% it gains barely 10%. A level transmitter reporting a linear 4-20 mA signal from a sphere, without characterisation, is wrong almost everywhere.
Worked example
A 3 m diameter LPG sphere filled to 1,500 mm — exactly half:
- r = 1.5 m, h = 1.5 m
- V = (π × 1.5² / 3) × (3 × 1.5 − 1.5) = (π × 2.25 / 3) × 3 = 7.069 m³
- Full sphere = (4/3)π × 1.5³ = 14.137 m³ — so exactly 50%, as symmetry requires
- At propane SG 0.51 that is about 3,605 kg of product
Field notes
- Spheres are never gauged by level alone in custody service. The non-linearity plus thermal expansion of the shell means serious inventory uses a certified strapping table with temperature correction, not a formula.
- LPG is stored as a liquid under its own vapour pressure, so the vapour space is pressurised and level is usually measured by differential pressure or servo gauge. Range a DP transmitter with the closed tank calculator, and note that LPG density changes markedly with temperature — a few percent over a summer day.
- Fill limits are regulated, not geometric. Pressurised LPG vessels are limited to a maximum fill (commonly around 85% by volume) so there is vapour space for thermal expansion. Capacity and safe fill are different numbers.
- Put characterisation breakpoints where the curve bends — tight near the top and bottom, sparse through the middle. Evenly spaced points waste resolution exactly where the error is worst.
- For cylindrical vessels with dished rather than hemispherical ends, use the dished head calculator; for cone bottoms, the cone bottom calculator.
Frequently asked questions
How do I calculate the volume of liquid in a spherical tank?
Use the spherical cap formula: V = (πh²/3)(3r − h), where r is the sphere radius and h the liquid depth from the bottom. At h = 2r it correctly returns the full sphere volume of (4/3)πr³.
Why are LPG storage tanks spherical?
A sphere has the smallest surface area for a given volume and carries pressure uniformly in every direction, so it needs the least steel for a pressurised duty. Smaller volumes use horizontal bullets, which are cheaper to fabricate but less material-efficient.
How do I calculate volume in a bullet tank with hemispherical ends?
Treat it as a horizontal cylinder plus one complete sphere, since two hemispherical heads make a sphere. Add the cylindrical segment area times length to the spherical cap volume at the same liquid depth.
Is level to volume linear in a spherical tank?
No, and it is the most non-linear common geometry. Near the bottom and top a small level change moves very little volume; near the equator the same change moves a great deal. Half the diameter is exactly half the volume, by symmetry.
Geometric calculation for reference and education; ignores wall thickness, internals and thermal effects, and is not a substitute for a certified strapping table or the applicable fill-limit regulations. See our disclaimer.