Cone Volume Calculator Explained: From Radius and Height to Every Other Measurement

What a cone is, and what its volume actually measures

A right circular cone is the simplest curved solid you can build from a circle: take a flat disc, pick a point on the axis above its centre — the apex — and join every point on the rim of the disc to the apex with a straight line. The result is a solid bounded by one flat circular base and one curved lateral surface that tapers smoothly to a single point. Two numbers fully describe it: the radius r of the base and the perpendicular height h from the centre of the base up to the apex. Feed those two into the cone volume calculator and you get five outputs — volume, slant height, diameter, base area, lateral surface area and total surface area — computed at full machine precision against Math.PI.

“Volume” here means the amount of three-dimensional space the cone encloses. If your radius and height are in centimetres, the volume comes out in cubic centimetres, which is the same as millilitres of water or ice cream the cone would hold. In metres, the volume is in cubic metres (1 m³ = 1,000 litres); in inches, in cubic inches (1 in³ ≈ 16.387 mL). The calculator never assumes a unit system — whatever linear unit you put in is the linear unit you get back, with areas squared and volumes cubed.

The single most useful fact about a cone is the one Archimedes proved in the 3rd century BC: a cone is exactly one third of the cylinder that would enclose it. Same base, same height, one-third the contents. Stack three identical paper cones inside a matching cylinder and they fit with no air left over. That 1/3 is exact, not an engineering approximation, and it is the reason the volume formula carries a leading factor of one third instead of one or one half.

The formulas the calculator uses

Every output the cone volume calculator prints is a direct function of r and h. There is no estimation, no lookup table; the math is closed-form and exact up to the precision of double-precision floating point.

Volume

V = (1/3) · π · r² · h

Slice the cone into thin horizontal discs perpendicular to its axis. At height z above the apex, where the cone has shrunk linearly from the full base radius r at z = h down to zero at z = 0, the disc radius is r(z) = r · (z / h) and its area is π · r(z)². Integrate from z = 0 to z = h:

V = ∫0h π · (r · z / h)² dz = (π · r² / h²) · [z³ / 3]0h = (1/3) · π · r² · h

That is one third of the matching cylinder’s volume π · r² · h. The same result falls out of Cavalieri’s principle without integration, which is how Eudoxus and Archimedes worked it out before calculus was invented.

Slant height

ℓ = √(r² + h²)

The slant height ℓ is the straight-line distance from any point on the rim of the base to the apex, measured along the curved surface (which is straight in the radial direction). It is the hypotenuse of a right triangle whose two legs are the radius and the perpendicular height — Pythagoras, applied to a cross-section that contains the axis. Slant height drives the lateral surface area; perpendicular height drives the volume. Mixing the two is the most common cone-arithmetic error, which is why the calculator displays both side by side.

Lateral and total surface area

Alat = π · r · ℓ    Atotal = π · r · (r + ℓ)

Cut the cone up the side from base to apex and unroll the curved face flat. It does not lie down as a triangle — it lies down as a sector of a circle of radius ℓ, with the arc of the sector being the original base circumference 2πr. The area of any circular sector is half its radius times its arc length, so A_lat = (1/2) · ℓ · 2πr = πrℓ. Add the circular base πr² for the total surface area, which factors neatly as πr(r + ℓ).

Diameter and base area

d = 2r    Abase = π · r²

Both are just the circle of the base. The diameter is twice the radius; the base area is the area of a flat circle of radius r. The circle calculator handles these stand-alone if you only need the base.

Worked example: the 3-4-5 cone

Take the default inputs in the cone volume calculator: r = 3 and h = 4. The slant height comes out to a clean integer, which is why textbook problems pick this combination again and again. Step through each output by hand to confirm what the calculator returns.

• Diameter: d = 2 · 3 = 6 units.
• Slant height: ℓ = √(3² + 4²) = √25 = 5 units — the classic Pythagorean triple.
• Base area: π · 3² = 9π ≈ 28.274334 square units.
• Lateral surface area: π · 3 · 5 = 15π ≈ 47.123890 square units.
• Total surface area: 9π + 15π = 24π ≈ 75.398224 square units.
• Volume: (1/3) · 9π · 4 = 12π ≈ 37.699112 cubic units.

Plug r = 3 cm and h = 4 cm into the calculator and the units snap into place. The cone has volume 37.7 cm³, which is 37.7 mL of water — a couple of tablespoons. Total surface area is 75.4 cm², about the area of a credit card and a half. Now scale up: r = 30 cm and h = 40 cm. The slant height grows tenfold (from 5 to 50), areas grow hundredfold (lateral surface is now 4,712 cm² ≈ 0.47 m²), and the volume grows a thousandfold (37,699 cm³, or 37.7 litres). That cube-square-linear scaling is the most useful intuition to carry around about cones, and we return to it below.

If your r and h land in inches and you want the volume in litres, run the volume converter afterwards — 1 in³ is about 16.387 millilitres, so a paper cup-cone with r = 1.5 in and h = 4 in has a volume of 3π ≈ 9.42 in³, or about 154 mL.

Where the cone formula shows up in real work

Ice-cream cones, paper cups, and party hats

A standard waffle cone is about 5 cm across at the rim and 12 cm tall, so r = 2.5 cm and h = 12 cm give a volume of (1/3) · π · 6.25 · 12 = 25π ≈ 78.5 cm³. That is the maximum amount of ice cream the cone can hold flush with the rim — the visible scoop on top is roughly hemispherical and contributes another (2/3) · π · 2.5³ ≈ 32.7 cm³, for around 111 cm³ total. The matching cylinder of the same r and h would hold three times the cone’s contents, which is why a cup-shaped tub of ice cream the same height holds three scoops to the cone’s one.

Funnels, hoppers, and grain silos

Industrial bulk-solid handling lives on cones. A conical hopper at the bottom of a silo gathers grain, sand, or powdered chemical down to a discharge gate; the cone’s steep walls keep the material flowing under gravity rather than bridging across a flat floor. Engineers size hoppers by volume, by slant angle (the angle between the wall and the vertical, which has to exceed the material’s angle of repose), and by surface area for the steel order. A typical farm grain hopper at r = 2.5 m and h = 3 m holds (1/3) · π · 6.25 · 3 ≈ 19.6 m³, or about 15.7 tonnes of wheat at 800 kg/m³.

Volcanoes, slag piles, and the angle of repose

Any granular material tipped onto a flat surface piles into a cone whose slope angle — the angle between the slant and the ground — matches the material’s angle of repose: about 34° for dry sand, 40° for gravel, 45° for damp clay. From the angle and the base radius you can recover the height with h = r · tan(angle), and then read the pile volume off the cone formula. Composite volcanoes like Mount Fuji are nearly perfect cones; Fuji’s base radius is about 18 km and its height 3.8 km, putting its total volume in the neighbourhood of 1,290 km³. The cone formula is the first approximation geologists use before they reach for digital elevation models.

Loudspeakers, rocket nozzles, and the unrolled sector

Anything that needs to flare smoothly from a small opening to a large one tends to be a cone or a near-cone. Loudspeaker cones are stamped from flat circular sectors of paper or polypropylene; the radius of the flat blank equals the slant height of the finished cone, and the arc length of the sector equals the rim circumference of the finished cone. Rocket bell nozzles start as cones in the textbook before being re-curved into ideal contours; the conical approximation already captures the thrust coefficient to within a few percent. The flat-sector identity A_lat = πrℓ is what lets the manufacturer cut the right blank without trial and error.

Filters, light cones, and viewing angles

Coffee filters, lab funnels, and acoustic horns all rely on a cone’s ability to concentrate or spread a flow. Optical and acoustic engineers describe directional fields using a cone of half-angle θ, and the solid angle the cone subtends is Ω = 2π · (1 − cosθ) steradians. That is a different number from the cone’s volume, but it builds on the same geometry — the intersection of an apex with a base circle of half-angle θ.

Cones compared with cylinders and spheres

The cone is the smallest of three classical solids that share the same bounding box. Sitting a cone, a sphere and a cylinder side by side, all with base radius r and height 2r, Archimedes’ great result is that their volumes are in the exact ratio 1 : 2 : 3.

• Cone (base r, height 2r): V = (1/3) · π · r² · 2r = (2/3) · π · r³.
• Sphere (radius r): V = (4/3) · π · r³.
• Cylinder (radius r, height 2r): V = π · r² · 2r = 2π · r³.

Two of these scoop-fills of the cone exactly equal the sphere; three of them exactly fill the cylinder. The sphere volume calculator and cylinder volume calculator cover the other two corners of the same trio, and any of the three results can be derived from the others using the 1:2:3 ratio without re-doing the integration.

How to reduce or increase cone volume in practice

Trade height for radius when volume matters more than footprint

Volume scales as r² but only linearly with h, so doubling the radius quadruples the contents while doubling the height only doubles them. When you have a height limit (a low ceiling, a road-haulage cap) but a free footprint, widen the cone. When you have a footprint limit (a narrow plot, a packaging carton) but free height, raise it. The same input numbers run through the cone volume calculator will tell you immediately which way the leverage points.

Trade slant for steepness when surface area matters

Steeper cones (large h/r ratio) have lateral surface much closer to πrh than to πr² — for a tall thin cone the lateral area grows almost linearly with height. For flat wide cones (small h/r ratio) the lateral surface is only a little more than the base area. Sheet-metal cost, heat-transfer area and material strength all scale with surface area, so the right aspect ratio depends on whether you are paying for the skin or for the contents.

Watch for the angle of repose when you cannot choose the slope

A pile of granular material does not let you pick its shape — the slope is fixed by the material. To increase the volume you can store on a fixed footprint, you have to add edge containment (silo walls, retaining berms) and let the cone become a frustum or a flat-topped pile. Pure cones only appear when the material is genuinely free-standing.

Convert units before the cube, not after

Convert linear dimensions into the system you actually want the answer in before running the formula. Calculating volume in m³ from radius in metres and then converting to litres (factor 1,000) is much less error-prone than computing in m³ and trying to talk about it in cm³ (factor 1,000,000). When the answer needs to be in a different volume unit entirely — gallons, fluid ounces, cubic feet — use the volume converter afterwards.

Subtract for hollow cones

A conical shell — like a paper cup or a funnel — has volume equal to the outer cone minus the inner cone (the air space its contents would occupy). For a thin-walled cone the difference is small and the lateral surface area is the more useful number. For thick castings (concrete cone footings, ballast piles) compute the two solids separately and subtract.

Common mistakes

Using slant height in the volume formula

Volume needs the perpendicular height h, not the slant height ℓ. Using ℓ in place of h inflates the answer by the factor ℓ / h = √(1 + (r/h)²), which is 25% wrong on the 3-4-5 cone (5/4) and 41% wrong on a 45° cone (√2). The calculator labels the inputs explicitly so there is no ambiguity, and the slant height appears in the breakdown so you can see them side by side.

Treating the diameter as the radius

Volume goes as r², so feeding a diameter into a radius field multiplies the volume by 4. A “10 cm cone” almost always means a cone 10 cm across at the base, so r = 5 cm. The cone volume calculator asks for the radius explicitly and shows the diameter in the breakdown to make the conversion visible.

Forgetting the one-third

The most embarrassing mistake in cone arithmetic is dropping the 1/3 and computing the cylinder volume instead. A 10 cm radius cone, 30 cm tall, has volume 1,000π cm³ ≈ 3,142 mL, not 3,000π cm³ ≈ 9,425 mL. The 1/3 is exact and is the entire reason the cone formula is worth memorising as a separate identity at all.

Mixing linear, area, and volume units

Radius in centimetres gives surface area in cm² and volume in cm³. Switch to metres and the volume number drops by a million, not a thousand. The same word (“centimetre”) hides three different multipliers depending on whether you square or cube it, and this is the source of an enormous fraction of student errors. Standardise on one linear unit before you start, and convert volumes at the end.

Assuming the formula works for oblique cones

The volume formula V = (1/3) · π · r² · h holds for an oblique cone as long as h is the perpendicular height between the base plane and the apex. The surface-area formulas, though, assume a right cone where the apex sits directly above the centre — for an oblique cone the slant height is no longer the same all the way around the rim, and πrℓ underestimates or misrepresents the actual lateral area. Use this calculator for right cones; oblique cones require integration or numerical methods.

When to look beyond the calculator

The cone volume calculator on this page assumes a right circular cone — circular base, apex directly above the centre, straight sides. Three situations need more than these closed-form formulas. Oblique cones (apex offset from the centre) still have the same volume but a different, asymmetric lateral surface that has to be integrated. Frustums (a cone with the apex sliced off, leaving a smaller circle on top) need the frustum formula V = (1/3) · π · h · (R² + Rr + r²) where R is the larger radius and r the smaller. Non-circular cones (elliptical bases, polygonal bases) need their own derivations — the 1/3 factor survives in all of them, but the πr² base area gets replaced by whatever the actual base shape is.

For engineering work where wall thickness, material strength or fluid dynamics matter — not just geometry — the cone formula is the starting point, not the answer. Pair it with material data, structural codes, or a CFD model as the application demands.

Frequently asked questions

What is the formula for the volume of a cone?

V = (1/3) · π · r² · h, where r is the radius of the circular base and h is the perpendicular height from the centre of the base to the apex. The factor of 1/3 makes the cone exactly one third of the cylinder with the same base and height.

How is slant height different from height?

Height (h) is the perpendicular distance from the centre of the base straight up to the apex. Slant height (ℓ) is the diagonal distance from the edge of the base to the apex along the curved side. They are connected by Pythagoras: ℓ = √(r² + h²). Volume uses h; lateral surface area uses ℓ.

How do I find the surface area of a cone?

Total surface area is πr(r + ℓ): the base πr² plus the lateral πrℓ. The cone volume calculator computes ℓ for you from r and h, so you only ever enter the two physical inputs.

Why is a cone exactly one third of a cylinder?

Integration of πr(z)² from z = 0 to z = h, with r(z) shrinking linearly from r down to 0, gives (1/3)πr²h. Archimedes proved the same result without calculus using Cavalieri’s principle. Three identical cones fit perfectly inside one matching cylinder; the 1/3 is exact, not an approximation.

How do I convert the cone’s volume into litres or gallons?

With r and h in centimetres, the volume is in cm³, which equals millilitres directly — divide by 1,000 for litres. In metres, the volume is in m³ (1 m³ = 1,000 litres). A US gallon is about 3.78541 litres; a UK gallon about 4.54609 litres. The volume converter handles any other pair.

Does the formula work for oblique cones?

The volume formula does, as long as h is the perpendicular height between the base plane and the apex (not the axial length along the tilted spine). The surface-area formulas assume a right cone, where the apex is directly above the centre; for an oblique cone the slant height varies around the rim and πrℓ no longer applies.

What if I know the slant height but not the perpendicular height?

Rearrange Pythagoras: h = √(ℓ² − r²). For r = 3 and ℓ = 5, h = √(25 − 9) = 4. Drop the recovered h into the cone volume calculator and the volume and surface areas follow as usual.

How does doubling the radius change the volume?

Volume is proportional to r², so doubling the radius multiplies the volume by 4 (with the same height). Doubling both radius and height multiplies the volume by 8 — the cone keeps its shape and the contents follow the cube of any linear scaling. The lateral surface area scales the same way as the base area, by a factor of 4.

Related calculators

• Cylinder Volume Calculator — the cone’s tripled counterpart; volume, lateral and base area of a right cylinder.
• Sphere Volume Calculator — the third member of the 1:2:3 Archimedean trio.
• Circle Calculator — radius, diameter, circumference, area of the flat base alone.
• Volume Converter — switch between cm³, m³, litres, gallons and ft³ once the cone has handed you its cubic answer.