A cone holds exactly one-third the volume of a cylinder with the same base radius and the same height. That is why the cone formula V = ⅓πr²h is just the cylinder formula V = πr²h with a ⅓ in front. Fill a cone with water and pour it into the matching cylinder: you need to do it exactly three times to fill the cylinder up.
It is one of those results that feels like it should be a half. The cone tapers steadily from a full circular base to a single point, so your instinct says the average cross-section is somewhere around half the base. The real answer is a third. And the reason why is genuinely elegant, not just a formula to memorise.
This guide walks through the intuition, a calculus proof, Cavalieri's principle, 2,400 years of history, practical uses, and five worked examples you can check against the Cone Volume Calculator.
Table of Contents
- The formula at a glance
- Why a third, not a half?
- The proof: calculus and discrete sums
- Cavalieri's principle: any cone, any tilt
- A 2,400-year-old discovery
- Worked examples
- The same rule for pyramids
- Practical uses of the one-third rule
- Related calculators
- Frequently asked questions
- References
The Formula at a Glance
A cone and a cylinder that share the same base radius r and height h:
| Shape | Volume formula |
|---|---|
| Cylinder | V = πr²h |
| Cone | V = ⅓πr²h |
| Ratio | Cone ÷ Cylinder = ⅓ |
Divide one formula by the other and the πr²h cancels completely, leaving exactly ⅓. The ratio holds for any size: a tiny ice cream cone, a full-size industrial silo, a sand pile in a field. As long as the two shapes share the same base and height, the cone is always a third.
Why a Third, Not a Half?
Here is the part most explanations skip.
Picture slicing the cone into thin horizontal discs, like a stack of coins that shrink toward the tip. As you move up the cone from base to apex, each disc gets narrower. The instinct says: if the radius shrinks steadily from full at the bottom to zero at the top, the average disc should be about half the base area. So the cone should be about half the cylinder.
But volume does not care about radius directly. It cares about the area of each disc, and area scales with the radius squared. That squaring changes everything.
Look at what happens halfway up. The disc there has half the radius of the base. But area scales with the square of the radius, so it has only ¼ of the base area (½ × ½ = ¼), not half. A disc three-quarters of the way up has only 9⁄16 of the base area (¾ × ¾ = 9⁄16). The upper portion of the cone contributes far less than intuition suggests, because its slices fade away quadratically.
Think of it this way: the cone spends most of its height with a very small cross-section. Add up all those squared-down slices across the whole height and the average cross-section comes out to exactly one-third of the base, not one-half. That single fact, area grows with the square of the radius, is the entire reason the answer is a third.
Here is a concrete numerical version. Split the cone into ten equal slices by height. The bottom slice (nearest the base) has 90% of the full radius, so its area is 0.9² = 81% of the base. The fifth slice (halfway up) has 50% of the radius: 0.5² = 25% of the base area. The ninth slice (near the tip) has 10% of the radius: 0.1² = just 1% of the base. Average those ten percentages and the result lands near 34%. The tiny upper slices pull the mean far below 50%, down toward a third. Taking infinitely thin slices and summing exactly gives 33.33…%.
The Proof: Calculus and Discrete Sums
If you have met integration, here is the clean version. Place the cone's apex at the origin and let a horizontal slice at height y have radius r × (y/h), because the radius grows in direct proportion to the height. The area of that slice is π × (ry/h)². Integrate from 0 to h:
The h³/3 in the middle is where the third comes from. It is the integral of y², the squared radius term. Same idea as before, made exact by calculus. No approximation, no hand-waving: the one-third drops out algebraically.
If you prefer to avoid integration entirely, the same result comes from a discrete sum. Divide the cone into n equal layers. Layer k from the apex has radius r×(k/n) and area πr²(k/n)², each with thickness h/n. Summing all n layers gives (πr²h/n³) × (1² + 2² + … + n²). The sum of squares equals n(n+1)(2n+1)/6, which approaches n³/3 as n grows. The total therefore approaches ⅓πr²h. You can get arbitrarily close with arithmetic alone. Calculus simply takes the limit cleanly in a single step.
Cavalieri's Principle: Any Cone, Any Tilt
Here is a neat extension. Cavalieri's principle says: if two solids have equal cross-sectional areas at every height, they have the same volume.
A slanted (oblique) cone has the same cross-sectional area at every height as an upright cone with the same base and height: you are just shifting each disc sideways, not resizing it. So the volumes are identical. A leaning cone holds the same as an upright one with the same base and height, and is still one-third of the matching cylinder.
The same logic applies to the cylinder comparison. The proof does not assume the cone is perfectly upright. It works for any cone, however it leans. And by the same principle, a cone with an elliptical base has the same volume as a cone with a circular base of equal area and equal height, so the one-third rule extends further still.
A 2,400-Year-Old Discovery
This is not a modern result. The Greek philosopher Democritus recognised the one-third relationship for cones and pyramids around 400 BC, though he could not yet prove it rigorously. A few decades later, Eudoxus of Cnidus proved it properly using the "method of exhaustion," an early forerunner of calculus. The proof appears formally in Euclid's Elements, Book XII, Proposition 10.
So every time you write V = ⅓πr²h on an exam or type it into a calculator, you are reaching for a result that took some of the sharpest minds of antiquity to establish.
Archimedes and the Perfect Triple
About a century after Eudoxus, Archimedes went further. He showed that when a sphere is inscribed inside a cylinder that exactly contains it (so the sphere's diameter equals both the cylinder's diameter and its height, meaning h = 2r), the three volumes fall into the simplest possible whole-number ratio:
| Shape | Formula (with h = 2r) | Ratio |
|---|---|---|
| Cone | ⅓πr²(2r) = 2⁄3πr³ | 1 |
| Sphere | 4⁄3πr³ | 2 |
| Cylinder | πr²(2r) = 2πr³ | 3 |
Cone : sphere : cylinder = 1 : 2 : 3. Every time the cone is one-third of the cylinder, the sphere is two-thirds of it, and the sphere holds exactly twice what the cone holds. Archimedes considered this the most beautiful result he had ever found. He reportedly asked for a diagram of a sphere inside a cylinder to be carved on his tomb, and the Roman statesman Cicero described finding the tomb identified by that very sculpture centuries later. The Sphere Volume Calculator gives (4/3)πr³ directly if you want to verify the 1:2:3 ratio for any specific r.
Worked Examples
Five shapes, all using V = ⅓πr²h. The Cone Volume Calculator verifies each of these in one step.
Example 1: Waffle Ice Cream Cone
Dimensions: radius 3 cm, height 10 cm.
V = ⅓ × π × 3² × 10 = ⅓ × π × 90 ≈ 94.2 cm³. The matching cylinder holds 282.7 cm³, exactly three times as much. This is why a generous scoop sits above the rim of the cone: the cone itself holds barely a third of what a cylinder of the same width would. You need three full cones to transfer one cylinder's worth of ice cream.
Example 2: Traffic Cone
Dimensions: base radius 15 cm, height 70 cm.
V = ⅓ × π × 15² × 70 = ⅓ × π × 15,750 ≈ 16,493 cm³ (about 16.5 litres). The matching cylinder would hold nearly 50 litres. Traffic cones are hollow and lightweight precisely because the conical shape uses so little material volume relative to the footprint they occupy. A solid cone of the same dimensions in rubber or plastic would use far more material than the external shape suggests.
Example 3: Party Hat or Funnel
Dimensions: radius 8 cm, height 25 cm.
V = ⅓ × π × 64 × 25 = ⅓ × 5,026.5 ≈ 1,675 cm³ (1.675 litres). A cylindrical container of the same width and height would hold 5,027 cm³: three times the volume for the same footprint. A kitchen funnel of similar dimensions empties quickly because that narrow lower section has minimal volume to drain, even if the funnel looks substantial from above.
Example 4: Conical Sand Pile
Dimensions: base radius 2 m, height 3 m.
V = ⅓ × π × 4 × 3 = ⅓ × 37.70 ≈ 12.57 m³ (12,566 litres). Useful for estimating aggregate, soil, or gravel stockpiled on site without measuring weight. The angle of repose for dry sand is roughly 34 degrees, so a 2 m base radius gives a natural height around 1.35 m. A 3 m height means the pile was either wet or manually shaped, and the formula still works regardless of how it formed.
Example 5: Working Backwards from Cylinder Volume
Given: a cylinder with r = 6 cm and h = 15 cm. What does the matching cone hold?
Cylinder volume = π × 36 × 15 = 1,696 cm³. Cone volume = 1,696 ÷ 3 ≈ 565 cm³. No need to use the cone formula directly: find the cylinder and divide by three. The Cylinder Volume Calculator gives the cylinder volume, then one extra division gives the cone. This shortcut works whenever you have the cylinder result already, such as when switching from a cylindrical container to a conical one of identical dimensions.
The Same Rule for Pyramids
The one-third relationship is not unique to cones. A pyramid is one-third of the prism (box) with the same base area and height, for the exact same reason: its cross-sections shrink with the square of the scale factor as you rise toward the apex.
A square pyramid with base side s and height h: V = ⅓ × s² × h. A rectangular pyramid with base l × w and height h: V = ⅓ × l × w × h. Both are one-third of their matching prism. The Pyramid Volume Calculator covers square, rectangular, and triangular bases.
In fact, any solid that tapers from a flat base to a single apex with cross-sections shrinking as the square of the height carries this same one-third factor. It is not a coincidence for circles; it is a universal consequence of the geometry of quadratic tapering. The Great Pyramid of Giza, a traffic cone, and a conical mountain spring all follow it, regardless of the shape of their base.
Practical Uses of the One-Third Rule
The cone formula appears in more everyday situations than most people realise.
Engineering and hoppers. Industrial hoppers, grain silos, and chemical reactors often terminate in a conical base. The conical section holds one-third of what the same cylinder would. Engineers factor this in when sizing storage: a hopper with a 1.5 m base radius and a 2 m conical section holds about 4.7 m³ in that section. Far easier to empty than a flat-bottomed cylinder, because gravity pulls material to the central outlet, but a third of the floor space wasted as a sloped surface you cannot fully fill.
Material stockpiles. Sand, gravel, coal, and grain pile into roughly conical shapes under gravity. Logistics teams use V = ⅓πr²h to estimate inventory volume from a single aerial photograph: measure the base radius and estimate height from shadow length, multiply, and divide by three. Environmental teams use the same technique to model overburden volumes on construction sites.
Food and portion control. A standard waffle cone (r = 3 cm, depth = 10 cm) holds 94 cm³. A small flat-bottomed cup of the same width and depth holds 283 cm³, three times as much. The one-third rule is the hidden geometry of every soft-serve stand and every funnel-cake booth at a fair.
Volcano and crater estimation. Field geologists routinely apply the cone formula to estimate the volume of cinder cones, crater rims, and tailings mounds from map measurements. The result is a lower bound for irregular shapes, but an order-of-magnitude estimate useful for hazard assessment and mine-closure planning without needing a survey drone.
Related Calculators
- Cone Volume Calculator: compute ⅓πr²h for any cone, with radius, height, and slant height options and full working steps.
- Cylinder Volume Calculator: the πr²h the cone is a third of, useful for comparing the two or working backwards from the cylinder result.
- Pyramid Volume Calculator: the same one-third rule applied to square, rectangular, and triangular pyramids.
- Sphere Volume Calculator: gives (4/3)πr³ and lets you verify the 1:2:3 Archimedes triple for any shared radius.
- Volume Calculators (all shapes): the full collection, including frustums, capsules, ellipsoids, and tanks.
Frequently Asked Questions
Is a cone exactly one-third of a cylinder?
Yes, exactly, as long as both shapes share the same base radius and height. The cone formula ⅓πr²h is precisely one-third of the cylinder formula πr²h: the πr²h part cancels and the ratio is ⅓ with no approximation.
Why is the volume of a cone one-third and not one-half?
Because volume is built from cross-sectional areas, and the area of each horizontal slice shrinks with the square of the radius. At the halfway point up the cone, the radius is half the base radius, so the area is ¼ of the base area, not ½. Those quadratically shrinking slices average out to exactly one-third of the base, not one-half.
How many cones of water does it take to fill the matching cylinder?
Exactly three. If the cone and cylinder have the same base radius and height, each cone holds one-third of the cylinder's capacity. This is a standard classroom demonstration: fill the cone, pour it into the cylinder, repeat. On the third pour the cylinder is precisely full.
Does the one-third rule change if the cone is slanted (oblique)?
No. By Cavalieri's principle, an oblique cone has the same cross-sectional area at each height as an upright cone with the same base and height. Equal areas at every height means equal total volume. The one-third ratio survives any amount of tilt.
Does the same one-third rule apply to pyramids?
Yes. A pyramid is one-third of the prism with the same base area and height. The cross-sections of a pyramid shrink quadratically toward the apex for the same geometric reason as a cone, so the one-third factor is the same. The formula is V = ⅓ × base area × height.
What is Cavalieri's principle and why does it matter here?
Cavalieri's principle states that if two solids have the same cross-sectional area at every height, they have the same volume. It matters for cones because it proves the one-third rule is not limited to perfectly upright cones. Any cone, however it leans, with the same base and height as a given cylinder holds exactly one-third of that cylinder's volume.
Who discovered that a cone is one-third of a cylinder?
Democritus is credited with first recognising the relationship around 400 BC. Eudoxus of Cnidus later provided a rigorous proof using the method of exhaustion, an early form of what we now call limits. The result appears formally in Euclid's Elements, Book XII, Proposition 10, making it one of the oldest proven volume results in mathematics.
How does integration prove the one-third?
Place the apex at the origin. A slice at height y has radius r × (y/h) and area π(ry/h)². Integrating from 0 to h gives (πr²/h²) × (h³/3) = ⅓πr²h. The factor of 3 in the denominator comes from integrating y², which gives y³/3. That is the algebraic origin of the one-third.
Does the one-third rule change if the cone is hollow?
The one-third rule describes the volume of the solid cone (or the capacity of a hollow cone of negligible wall thickness). A hollow cone with thick walls would need a different calculation. For a thin-walled hollow cone treated as a container, the interior volume is still ⅓πr²h where r and h are the interior dimensions.
What fraction of a sphere does a cone with the same base radius and height fill?
If the cone's height equals its base radius (h = r), then the cone volume is ⅓πr³ and the sphere of the same radius has volume (4/3)πr³. The ratio is (⅓) / (4/3) = ¼. So a cone with h = r fills one-quarter of a sphere with the same radius. For a general cone height, the ratio changes: compare the two formulas directly for any specific h and r.
How do I calculate a cone's volume if I only have the diameter?
Halve the diameter to get the radius, then apply V = ⅓πr²h. Or use the diameter form directly: V = ⅓ × π × (d/2)² × h = πd²h/12. Both give the same result. The Cone Volume Calculator accepts either diameter or radius as input.
References
- Wolfram MathWorld: Cone: derivation of the cone volume formula, geometric properties, and relationship to the cylinder.
- Wolfram MathWorld: Cavalieri's Principle: formal statement and proof of the principle used to extend the one-third rule to oblique cones.
- Britannica: Eudoxus of Cnidus: historical background on Eudoxus and the method of exhaustion used to prove the cone-cylinder ratio in antiquity.