Cone Volume Calculator
A cone holds exactly one-third of its cylinder. Enter the base radius and the vertical height, and this calculator returns the volume and the slant height — the pair every pile, funnel and roof calculation needs.
The Third Part
The cone's volume, V = (1/3)πr²h, is the cylinder's volume taxed by the famous third: same base, same height, a third of the contents. The third is universal for pointed solids (every pyramid takes the same third of its prism, a theorem the Egyptians used and Eudoxus proved), and its intuition is the tapering itself — the cone's slices shrink quadratically from base disc to point, averaging exactly a third of the cylinder's constant slice. Democritus knew the result a century before the proof, reportedly to his own puzzlement (if the slices are equal, it is a cylinder; if unequal, its surface is steps — the paradox that forced exhaustion's invention). The calculator pairs the volume with the slant height √(r² + h²) — Pythagoras up the cone's face — because the cone's uses (funnels, roofs, shades) are cut by the slant and filled by the volume, and a page that computes one owes the other.
Worked Example: Radius 3, Height 9
The defaults: V = π × 9 × 9 ÷ 3 = 27π = 84.82 cubic units — a third of the 254.47 the matching cylinder (the cylinder page at r = 3, h = 9) would hold, the third made tangible — and slant √(9 + 81) = √90 = 9.49. The lateral area by the note's recipe: π × 3 × 9.49 = 89.41, if a shade or funnel is being cut. Note the cone's personality in the numbers: at h = 3r it is a tall spike, and still the third holds — the ratio cares nothing for proportions, holding for a traffic cone and a volcano (a cinder cone's volume is genuinely estimated this way, from its base radius and height, by volcanologists pricing an eruption's output) alike.
The Volcano and the Lighthouse Beam
Two spectacles are cones in the wild: the cinder cone of the piles section, whose edifice a single eruption builds at repose — Parícutin's first-year cone (some 100 million cubic metres class) is this formula at geological manners — and the lighthouse beam, a cone of light whose cross-sections grow as r = distance × tan(beam angle), so its illuminated disc's area grows quadratically downrange while the volume of lit air between lamp and shore is a frustum of this page's solid. Both are estimated, in practice, by exactly the reader's two fields; nature and optics share the geometry because both are governed by straight lines from a point meeting a disc — the cone's definition, wearing weather in one case and photons in the other.
Piles, Hoppers and the Angle of Repose
Loose material builds cones for a living: a stockpile of gravel or grain settles at its angle of repose (typically 30–45° from horizontal, material-specific), which fixes h = r × tan(angle) and turns a footprint measurement into a tonnage estimate through this page — surveyors of inventory do exactly this flyover arithmetic. Hoppers and funnels are the inverted industrial form (frustums mostly, the cone's truncated sibling), and conical roofs and teepees wear the slant. The calculator's two fields are the pile survey's whole kit; the repose angle enters through the height field, converted by the reader's tangent, and its material-dependence is why the height is asked rather than the angle — the page computes geometry, the material's habit is the reader's local knowledge.
The Martini Glass Problem
The cone harbours a famous deception: fill a conical glass to half its HEIGHT and you have not half its volume but an eighth — the small cone at the top is similar to the whole at half scale, and volumes of similar solids cube their ratio (1/2)³ = 1/8 — so the top half of the height holds 1/8 of the drink and the bottom half the remaining 7/8. Bartenders, conical measuring flasks and chemistry's Erlenmeyer all exploit or suffer the same numerics: the level marks on a conical flask crowd near the top because equal height steps there add far less volume than at the base. The calculator cannot mark your flask, but entering its cone at two heights (full, and the top small cone at half) reproduces the 7:1 split in your own glassware's numbers — the most persuasive paragraph this page owns, and it is arithmetic.
Related Tools
More from the Math Calculators hub: the Cylinder Volume Calculator — the cone times three — and the Sphere Volume Calculator, Archimedes' other third-part relative.
Frequently Asked Questions
Why exactly one-third?
The deepest elementary answer is the slicing average: at a fraction t of the height from the apex, the slice's radius is t × r, its area t² × πr² — and the average of t² along the height is exactly 1/3 (the integral of t² from 0 to 1), so the volume is base × height × 1/3. Euclid's exhaustion proves it without calculus by inscribing ever-finer stepped pyramids; Cavalieri's principle makes it a one-liner (equal slice-areas, equal volumes, and a dissectible pyramid exhibits the third directly). It is one of mathematics' most re-proved facts, because every era insists on seeing the third with its own tools — now including this page's arithmetic.
What is a frustum, and how do I price one?
A cone with its tip cut off — the bucket, the hopper, the lampshade — whose volume is the difference of two cones: the whole cone minus the missing tip, computed by extending the frustum's sides to their apex (similar triangles give the full height from the two radii and the frustum's height). Two runs of relatives, then: this page prices whole cones, and the frustum's bridge is the subtraction, stated here because half the "cones" a reader meets in the wild are frustums wearing the shorter name — buckets especially, whose taper is small enough to forgive the approximation and large enough to matter to a concrete order.
Does the third hold for oblique cones?
Yes — Cavalieri again: lean the cone (apex no longer over the centre) and every horizontal slice keeps its area, so the volume stands at (1/3)πr²h with h the vertical height. The slant, though, loses its single value (the lean makes the face longer on one side), and with it the neat lateral formula — which is why the calculator assumes the right cone its diagram implies and says the slant belongs to that case. Piles and roofs are right cones by gravity's courtesy; volcanoes lean, and their pricing takes the volume and waives the slant.
How does the cone relate to the sphere and cylinder of Archimedes?
By the family portrait: for radius r and height 2r proportions, sphere : cone : cylinder volumes stand 4 : 2 : 6 in units of πr³/3 — the 2:3 sphere-to-cylinder ratio neighboured by the cone's own third — all three proved in the same treatise and all three living on this hub's adjacent pages. A reader who runs the trio at one radius owns the classical volumes entire: point, ball and can, the complete solid geometry of the ancient world in three widgets.
What is the lateral area, exactly, and why pi r slant?
Unroll the cone's face and it becomes a sector of a circle of radius equal to the slant, whose arc is the base's circumference 2πr: the sector's area is (arc ÷ full circle) × π × slant² = (2πr ÷ 2π × slant) × π × slant² ÷ … simplified, πr × slant — the note's recipe. It is the lampshade-pattern formula (the sector IS the pattern, cut flat) and the teepee canvas computation; the calculator prints the slant precisely so the reader can finish this one step, the last multiplication, with the geometry understood rather than a fourth formula memorised.
How is an ice-cream cone's capacity figured?
As this page prices it, once the cone's true inner radius and depth are measured — a standard wafer cone (r ≈ 2.5 cm, h ≈ 11 cm) holds about (1/3)π × 6.25 × 11 ≈ 72 cm³, roughly 72 ml, before the scoop's hemisphere (the sphere page's half) rides on top — and the parlour arithmetic of "how much is in the cone versus on it" resolves to: mostly on it. Manufacturers size scoops by number (a #16 disher is a sixteenth of a quart) while the cone's own capacity hides in this formula; the reader who runs both pages owns the whole dessert's geometry.
What does doubling the radius do to the pile?
Quadruples the volume if the height is free to follow the angle of repose (h scales with r, so V grows eightfold in truth — radius doubled at the SAME repose angle means height doubled, and r²h grows by 2³) — a distinction stockpile surveyors live by: a pile twice as wide is not four but eight times the tonnage, and a calculator run at the old height would understate the inventory by half. Enter the height the repose actually gives (the radius times the angle's tangent), and the page's figure is the surveyor's.
How tall is a cone from its volume and radius?
Invert the formula: h = 3V ÷ πr² — three times the volume spread over the base — the design direction (a silo cone, a pile limited by tonnage) as opposed to the survey direction. The calculator solves the forward problem; the inversion is printed because cone questions arrive in both directions with equal frequency, and the factor 3's position (numerator here, denominator there) is the detail every hurried inversion gets wrong. Check an inversion by re-entering the found height here: the round trip is the audit.