Two sections, two completely different curves
A cone bottom tank is a linear volume problem sitting on top of a cubic one, and the join between them is where level instrumentation gets interesting.
Inside the cone, liquid at depth y forms a smaller similar cone, so its volume scales with y³. Fill the cone to half its height and you have one eighth of the cone volume, not half. That non-linearity sits exactly where low-level alarms, pump protection and batch end-points live — which is why a cone bottom tank needs a characterisation curve far more urgently than a plain cylinder does.
Worked example
2 m diameter, 2 m straight shell, 800 mm full cone bottom, filled to 1,000 mm:
- Cone volume (full) = (1/3)π × 1² × 0.8 = 0.838 m³
- Cylinder volume = π × 1² × 2 = 6.283 m³ → total 7.121 m³
- At 1,000 mm the cone is full (800 mm) plus 200 mm of cylinder
- V = 0.838 + π × 1² × 0.2 = 0.838 + 0.628 = 1.466 m³ = 1,466 L
Note what the first 800 mm bought: only 0.838 m³, while the next 200 mm added 0.628 m³. Nearly the same volume from a quarter of the height.
Cone angle and the drainage reason
Cones exist to drain. A flat-bottomed tank leaves a heel of product that has to be pumped or washed out; a sloped base lets gravity finish the job. For solids and slurries the cone half-angle has to be steeper than the material's angle of repose, which is why hopper cones are often 60° included or steeper and why a shallow cone on a cohesive powder bridges and rat-holes.
| Included angle | Cone height on 2 m dia. | Typical use |
|---|---|---|
| 120° | 577 mm | free-flowing liquids |
| 90° | 1,000 mm | general process |
| 60° | 1,732 mm | slurries, crystallisers |
| 45° | 2,414 mm | cohesive powders |
Field notes
- A truncated cone is the real geometry. Every cone bottom ends in an outlet nozzle, so the true shape is a frustum. On a small tank with a large outlet the difference is material — enter the outlet diameter above rather than assuming a point.
- DP level measurement sees head, not volume. The transmitter reports level; the cubic-to-linear conversion happens in the control system. Range it with the open tank or closed tank calculator, and remember the reading depends on an assumed density.
- Put breakpoints where the curve bends. A characterisation curve needs tight spacing through the cone and can be sparse through the straight shell. Evenly spaced points waste resolution exactly where accuracy matters.
- Agitators, coils and baffles displace liquid that this geometric figure includes. A calculated capacity is an upper bound, not a strapping table.
- For dished and elliptical bottoms instead of a cone, use the dished head calculator; for plain cylinders, the vertical and horizontal tools.
Frequently asked questions
How do I calculate the volume of a cone bottom tank?
Add the cylinder and the cone: V = πr²·h_cyl + (1/3)πr²·h_cone. For a truncated cone with an outlet, the bottom section uses the frustum formula (1/3)πh(R² + Rr + r²).
How do I find the liquid volume at a given level in a cone bottom tank?
Inside the cone the liquid forms a smaller similar cone, so volume grows with the cube of depth — a quarter-full cone by depth holds only a sixty-fourth of the cone volume. Above the cone the relationship becomes linear.
What is the volume of a truncated cone or frustum?
V = (1/3)πh(R² + Rr + r²), where R is the larger radius, r the smaller and h the vertical height between them. It reduces to the plain cone formula when r is zero.
Why do hoppers and silos use conical bottoms?
A sloped base lets solids and viscous liquids drain completely under gravity rather than settling on a flat floor. The cone angle has to exceed the material angle of repose, which is why hopper cones are often steep.
Geometric calculation for reference and education; ignores wall thickness, nozzles and internals. Not a substitute for a certified strapping table. See our disclaimer.