A 2.5-inch-radius, 6-inch-tall sugar cone encloses (1/3)π × 2.5² × 6 ≈ 39.3 cubic inches of space. The cone's volume is exactly one-third of the cylinder with the same base and height (V = ⅓πr²h) — three cones of sand fill their cylinder precisely, a fact that delights every kid who tries it. The slanted side adds its own numbers: the slant height s = √(r² + h²) feeds the lateral area πrs, the paper the cone is rolled from. For the sugar cone, the slant height is exactly 6.5 inches, the lateral area is about 51.05 square inches, and the total surface is about 70.69 square inches — the three numbers a packaging engineer needs to size the cone, the wrapper, and the box it ships in. Traffic-cone and funnel manufacturers run the same three outputs.

The formula, and why one-third

A cone is a pyramid with a circular base, so it inherits the pyramid's one-third rule exactly. The base area is πr², the height is the perpendicular distance from base to apex, and the volume is one-third of their product. The cylinder with the same base and height holds πr²h, so the cone holds exactly a third of it, and the sand demo proves it in a minute: fill the cone, tip it into the cylinder, and three pours fill the cylinder to the brim. The one-third is not an approximation and it does not depend on the cone being tall or squat; it follows from the same quadratic taper that governs pyramids. Whenever a cone volume looks too big, check the division first: the un-divided product πr²h is the cylinder's volume, three times too big, and it triples every material order that follows from it.

Worked example: a kitchen funnel

A kitchen funnel with a top radius of 3 inches and a vertical height of 6 inches holds (1/3) × π × 9 × 6, about 56.55 cubic inches. In kitchen units that is 56.55 divided by 14.4375, about 3.92 US cups, and in metric it is 56.55 times 0.0163871, about 0.93 liters, just under a quart. The slant height is the square root of 9 plus 36, the square root of 45, about 6.71 inches, and the lateral area is π times 3 times 6.71, about 63.22 square inches of sheet metal if the funnel were unrolled flat. A cook scaling a recipe through this funnel cares about the cups figure; a manufacturer cares about the lateral area, because that is the blank size the press cuts. Both numbers come from the same two measurements, which is why the radius and height must be measured in the same unit before anything is multiplied.

Worked example: a party hat

A party hat with a radius of 4 inches and a height of 9 inches encloses (1/3) × π × 16 × 9, about 150.80 cubic inches of headroom, roughly 2.47 liters of air. The slant height is the square root of 16 plus 81, the square root of 97, about 9.85 inches, and that is the length of the hat's sloped edge, the measurement the elastic string must clear. The lateral area is π times 4 times 9.85, about 123.76 square inches of card, and adding the circular base of π times 16, about 50.27 square inches, gives a total surface of about 174.03 square inches. A hat maker buying card by the sheet divides the sheet area by 123.76, not by 174.03, because the base circle is usually a separate piece or left open; using the total surface for the card order overbuys by the base area on every hat. The volume, meanwhile, tells a different story: 150.80 cubic inches is about 0.65 US gallons, so the hat holds roughly two-thirds of a gallon of confetti if anyone is foolish enough to fill it.

Worked example: an ice cream cone and its scoop

A sugar cone with a radius of 2 inches and a height of 5 inches holds (1/3) × π × 4 × 5, about 20.94 cubic inches. A spherical scoop with a radius of 2 inches holds (4/3) × π × 8, about 33.51 cubic inches, so the cone plus one scoop totals about 54.45 cubic inches of dessert, roughly 0.89 liters. The comparison surprises most people: the scoop alone holds more than the cone it sits on, which is why a scoop overflows a cone that looks generously sized. A waffle cone with a radius of 2.5 inches and a height of 6 inches, the sugar-cone figure from the opening, holds 39.27 cubic inches, comfortably more than the 33.51-cubic-inch scoop, and that margin is the difference between a tidy serving and a melting mess. Ice cream shops size cones by this pair of numbers, not by looks, and the scoop-to-cone ratio is the quiet specification behind every menu photo.

Worked example: a gravel pile

A conical gravel pile with a radius of 9 feet and a height of 6 feet holds (1/3) × π × 81 × 6, about 508.94 cubic feet. Suppliers sell by the yard, so divide by 27: 508.94 divided by 27 is about 18.85 cubic yards, and ordering 19 yards covers it. The pile's shape is rarely a perfect cone — the top flattens where the loader drops and the base spreads where the dozer pushes — so experienced estimators add 10% and order 21 yards, then use the surplus on the next pile. The height here comes from the angle of repose: gravel poured from a fixed point naturally settles at about 33 to 35 degrees, so a 9-foot-radius pile cannot stand much taller than about 6 feet without mechanical help, and a pile quoted at 9 feet tall on a 9-foot radius should be questioned before it is paid for. Height, radius, and material angle are linked, and a cone volume that ignores the angle is a cone volume that ignores physics.

Running the formula backward

Often the volume is known and a dimension is missing. A cone must hold 100 cubic inches with a radius of 4 inches: the height is 3 times 100 divided by (π times 16), which is 300 divided by 50.27, about 5.97 inches. A traffic cone must hold 2,000 cubic inches with a height of 18 inches: the radius squared is 3 times 2,000 divided by (π times 18), which is 6,000 divided by 56.55, about 106.10, so the radius is about 10.30 inches. The backward forms are also the check on every forward answer: multiply π times radius squared times height, divide by 3, and compare with the claimed volume. If they disagree, the usual suspects are a diameter entered as a radius, which quadruples the answer, or a slant height used as vertical height, which inflates it by the slant-to-height ratio.

Scaling: radius squares, height only doubles

Cone volume scales with the square of the radius and linearly with the height, and the numbers prove it fast. The kitchen funnel at radius 3 and height 6 holds 56.55 cubic inches. Double the radius to 6, keeping height 6, and the volume is (1/3) × π × 36 × 6, about 226.19 cubic inches, four times as much. Double the height instead, radius 3 and height 12, and the volume is (1/3) × π × 9 × 12, about 113.10 cubic inches, only twice as much. This asymmetry is the estimator's friend and the buyer's trap: a funnel that looks "a bit wider" holds far more than it looks, while a taller funnel of the same width holds only proportionally more. The same law governs drill bits, rocket nose cones, and volcano models: widening pays quadratically, lengthening pays linearly.

The frustum trap: real funnels are cut off

A real funnel is almost never a full cone: its tip is cut off where the stem begins, and the shape is a frustum, a cone with its point removed. The frustum volume uses both radii: (1/3) × π × h × (R² + Rr + r²), where R is the top radius and r is the bottom radius. For a top radius of 3 inches, a bottom radius of 0.5 inches, and a height of 6 inches, that is (1/3) × π × 6 × (9 + 1.5 + 0.25), which is 2π × 10.75, about 67.54 cubic inches, against the full cone's 56.55. The frustum holds more because its wall stays further out over the lower section instead of tapering to a point. Using the cone formula for a real funnel therefore under-orders by about 16% in this case, and the error grows as the bottom radius grows. If the height you measured runs to where the point would be, below the stem, the cone formula overstates instead; the measurement decides the direction, so measure the actual sloped section, not the imagined point.

The diameter trap and honest measuring

Tape measures give diameters, not radii, and the cone formula wants the radius. A cone base measured across at 8 inches has a radius of 4 inches, and entering 8 as the radius quadruples the volume: π times 64 instead of π times 16. The same trap catches the height: the vertical height from base center to apex is the formula's height, while the slant height along the side is always longer and belongs only to surface-area math. Measure the base across, halve it, measure the height vertically with a square or a level, and only then multiply. A cone measured in inches gives cubic inches; a cone measured in centimeters gives cubic centimeters; mixing the two gives a number in no unit at all. The tool on this page takes radius and height together and returns volume, slant height, and total surface in one pass, so the measuring discipline is the only work left to the user.

Unit conversions that finish the job

Kitchen and shop units rarely match the formula's output, so the conversions belong with the formula. Cubic inches to milliliters: multiply by 16.3871, so the 56.55-cubic-inch funnel holds about 926.67 milliliters. Cubic inches to US gallons: divide by 231, so the funnel holds about 0.24 gallons and the party hat about 0.65 gallons. Cubic feet to cubic yards: divide by 27, so the gravel pile is 18.85 yards. Cubic centimeters to liters: divide by 1,000. The conversion that bites most often is cubic inches to cubic feet, dividing by 1,728: a cone quoted at 56.55 cubic inches is only 0.033 cubic feet, and a supplier who hears "56 cubic feet" instead of "56 cubic inches" will deliver a truckload for a thimble's job. State the unit with the number, every time, and the conversion errors mostly vanish.

How to use this calculator

Type the Radius (r) and Height (h) into their boxes and read the Volume, Slant Height (s), and Total Surface Area outputs together. The sugar cone shows about 39.3 cubic inches of volume, a 6.5-inch slant height, and 70.7 square inches of total surface — the three numbers a packaging engineer needs to size the cone, the wrapper, and the box it ships in. Traffic-cone and funnel manufacturers run the same three outputs. Use the vertical height, not the slant height, in the Height box — swapping them breaks the volume. And if the object is a real funnel with a stem, remember the frustum section above: this calculator's cone figure is the pointed-cone figure, and the frustum figure runs higher.

Five mistakes that cost real material

The first mistake is entering the diameter as the radius, which quadruples the volume. The second is using the slant height as the vertical height, which inflates it. The third is forgetting the divide-by-3, which triples it. The fourth is mixing units, inches with centimeters, which corrupts the product before any division happens. The fifth is pricing a real funnel as a full cone, which under-orders by the frustum gap, about 16% for the 3-inch funnel above. Run the cylinder check after every calculation: the cone must be exactly one-third of the cylinder with the same base and height, so a cone answer that exceeds its cylinder is wrong on its face, and a cone answer equal to its cylinder forgot the division.

Related calculators on this site

If the cone is one shape in a bigger job, three companion tools carry the next step. The volume of pyramid calculator handles the flat-sided cousin, where the base area is length times width and the same divide-by-3 applies. The volume of cylinder calculator sizes the enclosing cylinder, the three-times reference for any cone check. The slant height of cone calculator isolates the s = √(r² + h²) step whenever the wrapper or the roof edge is the goal, and the volume of sphere calculator sizes the scoop that sits on top of the cone. Each one uses the same discipline: radius, never diameter; vertical height, never slant; and the cylinder check before the order goes out.

A classroom note on the cone family

The cone belongs to a family of pointed solids that all share the one-third rule: pyramids with any polygonal base, cones with a circular base, and even oblique cones whose apex sits off-center, because the volume depends only on base area and perpendicular height, not on where the apex sits above the base. The sand demo proves it for the whole family: three cones fill their cylinder, three pyramids fill their prism, and the factor never wavers. That is why the cone formula survives every classroom demonstration and every factory floor: it is not an approximation tuned to ice cream, but a geometric fact about tapering, and the same fact that makes a volcano's volume estimable from its footprint and height alone.

With the pre-filled defaults — Radius (r) 3, Height (h) 5 — the Volume of Cone Calculator returns Volume 47.1239 m³, Slant Height (s) 5.8310 m, Total Surface Area 83.2298 m². Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.