About the Cone Calculator
A cone tapers from a circular base to a single point. Ice cream cones, party hats, funnels, spoil heaps, volcanoes, traffic cones, the pointed roof of a turret, the pile of sand a conveyor drops.
It has one fact worth knowing above all others, and one trap that catches almost everyone.
The fact: a cone holds exactly one third of the cylinder with the same base and height. Not approximately — exactly, and for every cone there has ever been.
The trap: a cone has two different heights, and the volume formula needs one while the surface area formula needs the other. Mixing them up gives a plausible answer that is wrong, which is the worst kind.
This calculator computes both heights whatever you give it, labels which is which, and shows what the wrong one would have produced.
How to Use the Cone Calculator
From the radius and height. The vertical height — straight up from the centre of the base to the tip.
From the radius and slant height. If you measured up the sloping side instead, start here and the vertical height is recovered for you.
From the diameter and height. For cones specified across the base.
Height needed for a volume. For sizing a funnel or a hopper to a capacity.
Two Heights, Two Formulas
h = vertical height centre of the base, straight up to the tip
l = slant height rim to tip, up the sloping surface
r = base radius
They form a right triangle: l = √(r² + h²)The slant is the hypotenuse, so it is always the longer of the two. If your slant is shorter than your height, one of them is mismeasured.
Now the formulas, with the height each one needs:
That split is the whole difficulty of cone problems. The volume is about how much space is underneath the surface, which depends on how tall the cone stands. The surface area is about the sloping skin itself, which depends on how far it is up the slope.
Use h where l belongs and the surface comes out too small — for a 3-4-5 cone, π×3×4 = 37.7 instead of π×3×5 = 47.1, a 20% shortfall. That is a real amount of material.
Why a Third of the Cylinder
Fill a cone with water and pour it into a cylinder of the same base and height. It takes exactly three cone-loads to fill. It is a classic demonstration, and it is genuinely exact.
The reason: measure down from the tip, and at distance x the cross-section is a circle whose radius grows in proportion to x. The area therefore grows as x², so the cone's cross-section is small near the tip for most of its length. Integrating x² from 0 to h gives h³/3, and the 3 in the denominator is where the third comes from.
The same holds for any pyramid or cone, whatever the base shape: one third of the prism on the same base and height. A square pyramid is a third of its box.
A practical consequence: a conical container has to be three times as tall as a cylindrical one on the same base to hold the same amount. Cone-shaped vessels look much larger than their capacity.
The Unrolled Cone
Why is the curved surface πrl?
Cut a cone up one side and flatten it. What you get is not a triangle — it is a sector of a circle whose radius is the slant height l. The sector's curved edge was the base circumference, 2πr.
A full circle of radius l would have circumference 2πl and area πl². Our sector has only 2πr of that edge, so it is the fraction r/l of a full circle:
curved area = π l² × (r / l) = π r lWhich is also how you mark out a cone in sheet metal or paper: draw a circle of radius l, cut out a sector covering r/l of it, and roll it up.
Step-by-Step Example
A cone with base radius 3 and height 4.
The cross-check: the cylinder on this base and height holds π × 9 × 4 = 113.097, and a third of that is 37.699. ✓
Note the 3-4-5 triangle doing the work again — the radius, the height and the slant of a cone are always a right triangle.
A conical heap of sand. A conveyor drops sand into a pile 6 m across at the base and 2.2 m high.
About 21 cubic metres — roughly 33 tonnes of dry sand.
Working backwards. A funnel must hold 250 cm³ with a 5 cm radius.
h = 3 × 250 / (π × 25) = 750 / 78.54 = 9.55 cmCompare: a cylinder of the same radius holding 250 cm³ would be only 3.18 cm tall. Three times shorter, exactly as expected.
Understanding Your Result
The volume is in cubic units.
The dimensions list the radius, the vertical height and the slant height together, explicitly labelled, so there is no ambiguity about which is which.
The total surface area includes the base.
The curved side and base are separated, because an open cone — a party hat, a funnel, a lampshade — needs only the curved part.
The slope gives the angle the side makes with the base, and the half-angle at the tip. These always add to 90°. The base angle is what a spoil heap's angle of repose means, and what a roof pitch would be.
When Should You Use This Calculator?
Heaps and stockpiles. Sand, gravel, grain and compost all pile into cones, and the volume gives the quantity from two easy measurements.
Funnels and hoppers. Capacity, and how tall a funnel must be for a given throughput.
Sheet metal and paper craft. The unrolled sector is the flat pattern to cut.
Roofing. A conical turret or spire needs the curved surface area for its covering, which uses the slant height.
Traffic cones, marquees and tepees. Material is the curved surface; enclosed space is the volume.
Ice cream and catering. How much a cone holds, and how many from a batch.
Volcanoes and geology. Cinder cones are modelled this way for eruption volumes.
Homework. Cone volume and surface area are standard exercises, and the slant height step is exactly where marks are lost.
Common Mistakes
Using the slant height in the volume formula. It gives an answer that is too large. The volume wants the vertical height.
Using the vertical height in the surface formula. It gives an answer that is too small. The surface wants the slant.
Using the diameter as the radius. Since the radius is squared, this gives four times the true volume.
Including the base when the cone is open. A party hat, a funnel and a lampshade have no base. Take the curved side only.
Assuming a tall cone holds a lot. It holds a third of what its cylinder would. Conical containers are much smaller than they look.
Measuring a heap's height at the edge. The height is at the centre, under the peak — the tallest point, not the average.
Forgetting that the slant is the hypotenuse. If someone gives you a "height" that is longer than the distance from centre to rim plus a bit, check which one they measured. The slant exceeds both the radius and the vertical height, always.