Cone Volume Calculator
Exactly one third of its cylinder, always.
Work out Cone Volume. Exactly one third of its cylinder, always. Every convention shown, not just one.
Volume (cubic units)
37.699112
Surface area 75.3982
A cone is exactly one third of the cylinder that contains it — same base, same height. Archimedes proved this without calculus, and the same one-third relationship holds for any pyramid over any base, which is a far more general result than it first appears.
How the Cone Volume Calculator works
Volume, surface area and slant height of a cone, with the enclosing cylinder alongside. A cone is exactly one third of that cylinder — a result Archimedes proved without calculus.
Also known as: volume of a cone · cone slant height calculator · why is a cone a third of a cylinder · conical hopper volume
Exactly one third, and why it generalises
A cone is precisely one third of the cylinder with the same base and height. Archimedes proved it geometrically; integration confirms it by summing discs whose radius shrinks linearly from base to apex.
The same one-third holds for any pyramid over any base — square, triangular, irregular. It is a far more general result than the cone case alone suggests, and it follows from the linear taper rather than from the base shape.
It also survives shearing. An oblique cone, whose apex is not above the base centre, has the same volume as the upright one of equal base and perpendicular height. That is Cavalieri's principle, and it is why the formula makes no mention of tilt.
Slant height and the flat pattern
Slant height is the hypotenuse of the radius and the vertical height, so a cone of radius 3 and height 4 has a slant of exactly 5. It is not the same as the height, and confusing them is the most common cone error.
The curved surface unrolls into a sector of a circle whose radius is the slant height. The sector's arc must equal the cone's base circumference, which fixes the sector angle and gives the flat pattern for making one.
That is why the curved surface area is πrl rather than anything involving the vertical height. The formula is describing the flattened sector, not the standing cone.
Frustums and the angle of repose
A frustum is a cone with the top cut off parallel to the base — the shape of most buckets, plant pots and lampshades. Its volume is the full cone minus the removed tip, or directly πh(R² + Rr + r²) ÷ 3.
Conical stockpiles have their own constraint: the angle of repose, the steepest slope a loose material will hold. For dry sand it is around 30 to 35 degrees, and it fixes the height for any base diameter.
That makes stockpile volume calculable from the base alone, which matters because measuring the height of a large pile is impractical. The angle substitutes for a measurement that cannot easily be taken.
Where to go next
The Cone Volume question rarely arrives on its own. These are the ones that usually come with it:
- Cylinder Volume Calculator — Widening beats heightening — radius is squared.
- Sphere Volume Calculator — Two thirds of its cylinder — Archimedes' favourite result.
- Polygon Area Calculator — Exterior angles always sum to 360°, whatever the shape.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
What is the formula for cone volume?
πr²h ÷ 3 — one third of the cylinder with the same base and height. The same one-third relationship holds for any pyramid over any base.
What is slant height?
The distance from the apex down the side to the rim, found by Pythagoras from the radius and vertical height. A cone with radius 3 and height 4 has a slant height of exactly 5.
How do I find the surface area?
πr(r + l), where l is the slant height. That is the circular base plus the curved surface, which unrolls into a sector of a circle.
Why is a cone one third of a cylinder?
Archimedes proved it geometrically; calculus gives it by integration. It is the same one-third that applies to every pyramid, which is a more general result than the cone case alone suggests.
What is a frustum?
A cone with the top cut off parallel to the base — the shape of most buckets and lampshades. Its volume is the difference between the full cone and the removed tip.
How do I make a cone from flat material?
Cut a sector of a circle with radius equal to the slant height. The arc length of the sector must equal the circumference of the cone's base, which sets the sector's angle.
How do I find the volume of a frustum?
Subtract the removed tip cone from the full cone, or use the direct formula πh(R² + Rr + r²) ÷ 3. It is the shape of most buckets, lampshades and plant pots.
What is the angle of repose?
The steepest angle a loose material forms when poured into a cone. It varies by material — around 30 to 35 degrees for dry sand — and it determines how much a stockpile of a given base holds.
How do I calculate a conical stockpile?
Use the base radius and the height implied by the angle of repose. Measuring the height directly on a large pile is impractical, so the angle is used instead.
Why is the lateral surface a sector?
Because unrolling the cone's curved surface flattens it into a sector of a circle whose radius is the slant height. The sector's arc must equal the base circumference, which fixes its angle.
What is an oblique cone?
One whose apex is not above the centre of the base. Its volume formula is unchanged — still a third of base times perpendicular height — which is a consequence of Cavalieri's principle.
How does a cone compare with a pyramid?
The same one-third relationship holds for both, over any base shape. A cone is simply a pyramid with a circular base, and the volume formula treats them identically.
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