How this calculation works
Use two parallel circular ends of a right conical frustum. Derive slant length from the perpendicular height, then keep lateral area separate from the two disk areas.
V = πh(r1²+r1r2+r2²)/3; slant = √(h²+(r1−r2)²); lateral area = π(r1+r2) × slantA worked example
End radii 4 m and 2 m with perpendicular height 3 m give volume 28π ≈ 87.964594 m³. Slant length is √13 m; curved area is 6π√13 m² and the two disk areas add 20π m².
Good to know
Right circular frustum only. One end can reduce to a point; both radii cannot be zero. Equal radii give the cylinder limit. Closed area includes both ends; wall thickness, internal capacity and waste are not inferred.
Reference: MathWorld: geometry definitions and formulas (opens in a new tab) ↗
Perpendicular height and slant length belong to different formulas
With end radii 4 m and 2 m and perpendicular height 3 m, the side slant is √(3²+(4−2)²) = √13 m. The volume uses the 3 m perpendicular distance between the circular end planes, not that longer side measure.
Volume is π × 3 × (16+8+4)/3 = 28π, about 87.964594 m³. Using an average radius first and applying a cylinder formula would give a different result: radius changes along the height, and cross-sectional area depends on radius squared.
The cylinder calculator checks the equal-radius limit. The cone volume calculator checks the zero-radius end limit; both are special cases of the stated frustum model.
Curved material and end closures are separate surfaces
The curved area is π(4+2)√13 = 6π√13 m². The circular end disks add 16π and 4π m². The reported closed surface includes both; an open-ended object would need only the surfaces actually present.
Those ideal areas do not include wall thickness, a seam, overlap, waste or a distinct internal cavity. Changing only perpendicular height changes slant length nonlinearly even though volume changes linearly. That is why the height comparison does not simply scale every surface entry by the same factor.
Cone and cylinder limits provide useful checks
Equal end radii turn the shape into a cylinder, with slant equal to perpendicular height. Reducing one end radius to zero gives a cone and one zero-area end disk. Swapping the two radii does not change volume or curved area.
The cylinder calculator and cone volume calculator cover those separate special cases. This frustum page needs two end radii because both can be positive and unequal. It is a right circular model with parallel end planes, not a method for an oblique or irregular tapered container.
A few questions, answered
Can the smaller radius be entered first?
Yes. The volume and curved-area expressions are symmetric in the two radii.
Does closed area assume both ends exist?
Yes. End disk areas are also shown separately so the boundary convention stays visible.
Can one radius be zero?
Yes, giving the cone limit. Both cannot be zero.
Does this calculate a hollow container’s capacity?
Not unless the dimensions already describe the intended internal geometry. No wall thickness is inferred.