Cone Volume Calculator

The right circular cone is one of the most familiar shapes in three-dimensional geometry. It has two defining measurements, the radius r of the circular base, and the perpendicular height h from the base to the apex, and from those two numbers you can derive the entire shape: the volume V = (1/3)·π·r²·h, the slant height l = √(r² + h²), the base area π·r², the lateral area π·r·l, and the total surface area π·r² + π·r·l. This calculator takes the radius and the height as input and reports every one of those numbers instantly, with the precision of a standard IEEE 754 double. It also runs the calculation backwards: enter a known volume together with a known height and the tool recovers the radius, or enter a known volume together with a known radius and the tool recovers the height; in either reverse case the slant height, base area, lateral area, and total area are all recomputed and reported in the same pass.

Cones appear almost everywhere. Ice cream cones hold a single scoop of dairy or oat. Traffic cones mark lane closures on motorways. Party hats taper from a circular brim to a tip in coloured card. Funnels transfer liquid or powder between containers of different sizes. Chemistry uses conical Erlenmeyer flasks for titration and conical centrifuge tubes for sample prep. Architecture uses conical roofs for circular buildings and conical tents for outdoor events. Manufacturing uses conical hoppers to feed material from storage into a downstream process. Even geology approximates volcanoes and certain mountain peaks as cones for first-order volume estimates. Anywhere the shape tapers from a circular base to a single point, this is the calculation that gives you the answer.

The classical derivation of the cone's volume is due to Archimedes of Syracuse, around 225 BCE, in his treatise On the Sphere and Cylinder. He proved (by mechanical argument and the method of exhaustion) that any right circular cone occupies exactly one-third of its circumscribing cylinder, the cylinder that has the cone's base as its base and the cone's height as its height. That single geometric fact is the entire algebraic content of the formula V = (1/3)·π·r²·h: the cylinder's volume is π·r²·h, and the cone's is one-third of that. The slant height l = √(r² + h²) is the Pythagorean theorem applied to the right triangle formed by the radius, the height, and the slant. The lateral area π·r·l is the perimeter of the base times the slant divided by two (the sector unrolling), and the total area is the lateral plus the base. All of these are exact to the precision of the inputs.

How to Use the Cone Volume Calculator

1. Pick a mode at the top of the tool: Forward (radius r and height h → volume V, slant l, base area, lateral area, total area), Reverse from volume and height (V + h → r → all of the above), or Reverse from volume and radius (V + r → h → all of the above).
2. In Forward mode, type the radius r and the height h into the two input fields. Any positive numbers work, with or without units; the answer cards assume the same length unit for r, h, and l, the same cubic unit for V, and the same square unit for the three areas. The output refreshes as you type.
3. In Reverse from volume and height, type a known volume V and a known height h. The calculator applies r = √(3V / (π·h)) to recover the radius and then reports the slant, the base area, the lateral area, and the total area. Useful for sizing a conical tank or hopper of a known depth that must hold a known volume.
4. In Reverse from volume and radius, type a known volume V and a known radius r. The calculator applies h = 3V / (π·r²) to recover the height and then reports the slant, the base area, the lateral area, and the total area. Useful for sizing a conical funnel or cup of fixed mouth diameter to a target capacity.
5. Click any preset in forward mode to populate the inputs with a unit cone, a 3-4-5 cone, a 5-12-13 cone, an equal-radius-and-height cone, a party hat, a traffic cone, a conical funnel, or a large hopper.

The answer cards show the radius r, the height h, the volume V, the slant height l, the base area π·r², the lateral area π·r·l, the total area π·r² + π·r·l, and the dimensionless ratio V / (r²·h), which is always exactly π/3 ≈ 1.04720. That last ratio is the easiest way to spot an algebraic slip: if it doesn't match π/3 to machine precision, the formula or the input is wrong.

The Formula

The classical right-circular-cone formulas are

Volume: V = (1/3)·π·r²·h

Base area: A_base = π·r²

Slant height: l = √(r² + h²)

Lateral area: A_lateral = π·r·l

Total surface area: A_total = π·r² + π·r·l

where V is the volume enclosed by the cone, A_base is the area of the circular base, A_lateral is the area of the curved lateral surface (the cone's "side"), A_total is the area of the entire outer boundary (base plus lateral), r is the radius of the base, h is the perpendicular height from the base to the apex, and l is the slant height measured from any point on the rim of the base to the apex along the cone's surface. The factor π is the universal circle constant, approximately 3.141592653589793. The factor 1/3 in the volume formula is the geometric statement that the cone occupies one-third of its circumscribing cylinder, Archimedes' classical result.

Reverse applications of the formulas are obtained by elementary algebra. To recover the radius from a known volume V and a known height h, solve V = (1/3)·π·r²·h for r² and take the positive square root:

Radius from V and h: r = √(3V / (π·h))

To recover the height from a known volume V and a known radius r, solve V = (1/3)·π·r²·h for h:

Height from V and r: h = 3V / (π·r²)

Both are well-defined for any positive inputs and produce positive outputs. The reverse calculations are mathematically exact (no iterative approximation is needed); the only rounding is in the display layer. This means that the precision of your reverse answer is limited only by the precision of the original measurement.

Worked Examples

Example 1, Unit cone (r = 1, h = 1)

The simplest non-trivial cone has equal radius and height, both equal to 1 in some length unit.

V = (1/3)·π·1²·1 = π/3 ≈ 1.04720 l = √(1 + 1) = √2 ≈ 1.41421 A_base = π ≈ 3.14159 A_lateral = π·√2 ≈ 4.44288 A_total = π + π·√2 ≈ 7.58448

This is the canonical reference object for cone geometry, the way the unit sphere (r = 1) is the reference object for sphere geometry. The volume is π/3 ≈ 1.04720, exactly one-third of the unit cylinder's volume of π ≈ 3.14159.

Example 2, 3-4-5 cone (r = 3, h = 4)

A Pythagorean cone with base radius 3 and height 4 has slant height exactly 5 (the famous 3-4-5 right triangle).

V = (1/3)·π·9·4 = 12·π ≈ 37.69911 l = √(9 + 16) = √25 = 5 = 5.00000 A_base = 9·π ≈ 28.27433 A_lateral = π·3·5 = 15·π ≈ 47.12389 A_total = 9·π + 15·π = 24·π ≈ 75.39822

Notice how clean the algebra is when r, h, l form a Pythagorean triple: the slant comes out exactly, and the volumes and areas are integer multiples of π.

Example 3, 5-12-13 cone (r = 5, h = 12)

A larger Pythagorean cone with base radius 5 and height 12 has slant height exactly 13 (the 5-12-13 right triangle).

V = (1/3)·π·25·12 = 100·π ≈ 314.15927 l = √(25 + 144) = √169 = 13 = 13.00000 A_base = 25·π ≈ 78.53982 A_lateral = π·5·13 = 65·π ≈ 204.20352 A_total = 25·π + 65·π = 90·π ≈ 282.74334

Volume ≈ 314.16 cubic units. This cone has exactly 100 times the volume of the unit cone, because the cross-sectional area is 25× and the height is 12×, giving 25 × 12 / 3 = 100 times the unit cone volume.

Example 4, Equal r and h (r = 10, h = 10)

A larger cone where the radius and the height are both 10, useful when the shape has the same footprint diameter (20) as its axial height.

V = (1/3)·π·100·10 = (1000/3)·π ≈ 1047.19755 l = √(100 + 100) = √200 = 10·√2 ≈ 14.14214 A_base = 100·π ≈ 314.15927 A_lateral = π·10·10·√2 = 100·√2·π ≈ 444.28793 A_total = 100·π + 100·√2·π ≈ 758.44720

Volume ≈ 1047.20 cubic units. The slant is 10·√2 because the right triangle is isoceles (r = h = 10), so the slant is just r·√2.

Example 5, Party hat (r = 0.1, h = 0.3 m)

A conical party hat with base radius 10 cm and height 30 cm (in metres: r = 0.1, h = 0.3).

V = (1/3)·π·0.01·0.3 = 0.001·π ≈ 0.00314 m³ = 3.14 litres l = √(0.01 + 0.09) = √0.1 ≈ 0.31623 m A_base = π·0.01 ≈ 0.03142 m² A_lateral = π·0.1·0.31623 ≈ 0.09935 m² A_total = 0.03142 + 0.09935 ≈ 0.13077 m²

A party hat holds about 3.14 litres if you filled it to the brim, which is more than enough for a generous scoop of whipped cream. The slant of about 31.6 cm is the length of card you'd need to wrap around the cone from the brim to the apex (minus the overlap).

Example 6, Reverse from volume and height (V = 100, h = 10)

A conical tank must hold 100 cubic units of fluid and is 10 units tall. What radius must its base be?

r = √(3·100 / (π·10)) = √(30 / π) ≈ √9.54930 ≈ 3.09017

So the base radius is approximately 3.09017 units (diameter ≈ 6.18). The slant is then l = √(3.09017² + 10²) ≈ √(9.54915 + 100) ≈ √109.54915 ≈ 10.46586, the lateral area is π·3.09017·10.46586 ≈ 101.59, and the total surface area is π·3.09017² + 101.59 ≈ 30.00 + 101.59 ≈ 131.59 square units. This is the area you'd have to coat, paint, or insulate.

Example 7, Reverse from volume and radius (V = 100, r = 5)

A conical funnel with a mouth radius of 5 units must deliver 100 cubic units of liquid per pour. How tall must it be?

h = 3·100 / (π·25) = 300 / (25π) = 12 / π ≈ 3.81972

So the funnel must be approximately 3.81972 units tall. The slant is then l = √(25 + 14.58776) ≈ √39.58776 ≈ 6.29109, the lateral area is π·5·6.29109 ≈ 98.81, and the total surface area is 25·π + 98.81 ≈ 78.54 + 98.81 ≈ 177.35 square units.

Where It Shows Up

The cone volume and surface area formulas appear in more real-world settings than almost any other pair in elementary geometry. Food and drink uses cones for ice cream cones, wafer cones, conical pastry shells, conical coffee filters (V60-style drippers), conical measuring cups, conical tasting spoons, and the cone-shaped dimples on a strawberry. Road safety and traffic uses cones for traffic cones, safety cones, pylon cones, and the orange cones that mark motorway lane closures. Events and parties use cones for party hats, conical party bags, conical gift wrap, and conical noisemakers. Cooking and chemistry use cones for conical flasks (Erlenmeyer), conical centrifuge tubes, conical filter funnels, conical separating funnels, conical Buchner funnels, conical pastry bags, and conical measuring jugs. Manufacturing and process engineering use cones for conical hoppers, conical feeders, conical mixers, conical screw conveyors, conical silos, and conical dust collectors. Architecture and construction use cones for conical roofs on circular buildings (the cone-on-cylinder gazebo, the circular temple, the oast-house kiln roof), conical tents (teepees, bell tents, conical marquees), conical acoustic horns in old gramophones and PA systems, conical traffic calming bumps, conical piles of sand, gravel, salt, or grain in stockpiles, and conical snow piles after plowing. Nature and geology approximate volcanoes, certain mountain peaks, sand dunes, alluvial fans, cones of depression in wells, and the conical settling tanks in water treatment as cones for first-order volume estimates. Medicine and biology use cones for the conical structure of the ear canal, the conical tip of a syringe, the conical cavity of certain tooth preparations, the conical section in CT imaging (less common but used in dental CBCT), and the conical cap of a joint replacement. Anywhere the shape tapers from a circular base to a single point, this is the calculation that gives you the answer.

Common Mistakes

Mistake 1, Confusing the right circular cone with the oblique cone. The formulas V = (1/3)·π·r²·h, A_lateral = π·r·l, and A_total = π·r² + π·r·l are valid only for a right circular cone, where the apex is directly above the centre of the base. For an oblique cone (apex shifted off-centre), the volume formula is still V = (1/3)·π·r²·h where h is the perpendicular height, but the lateral area is no longer π·r·l with l the straight-line slant; it requires the more general A_lateral = π·r·s where s is the slant along the surface, which varies around the rim. The calculator is for right circular cones; if your apex is off-centre, the answer will be slightly off.

Mistake 2, Mixing units across the calculation. If the radius is in metres, the volume is in cubic metres (m³), the height is in metres, the slant is in metres, and the three areas are in square metres (m²). A 1 m radius and 1 m height gives V ≈ 1.04720 m³ ≈ 1047.20 litres. If you typed 100 instead of 1 thinking it was in centimetres, you'd be off by a factor of a million on the volume. The simplest defence is to pick one unit, convert everything to that unit, and read the answer in the matching cubic and square units. Conical tank volumes are often quoted in litres; convert to cubic metres (1 L = 0.001 m³) before computing, or compute in cubic metres and multiply by 1000 to get litres.

Mistake 3, Confusing radius and diameter. The number that drops out of a measurement is usually the diameter (a ruler across the middle), but the formulas take the radius, which is half that. Forgetting this is by far the most common error and gives a cone whose volume is wrong by a factor of 4 (r²), lateral area wrong by a factor of 2 (r), and total area wrong by a mix of factors. If your ice cream cone spec says "5 cm across the top," that almost certainly means 5 cm diameter and therefore r = 2.5 cm.

Mistake 4, Approximating π as 22/7 or 3.14. For schoolbook homework that's fine, but for any engineering or scientific use, use the full π ≈ 3.14159265358979. The difference between 22/7 and the true value is about 0.04%, which on a 100-litre tank is 0.04 litres, sometimes that matters, sometimes it doesn't. The calculator uses the IEEE 754 double-precision value of π and so does not introduce any rounding error from π.

Mistake 5, Treating the hemisphere as a cone. A hemisphere is half a sphere, not a cone. The two formulas share some features (both involve π times a power of a radius), but V_hemisphere = (2/3)·π·r³ while V_cone = (1/3)·π·r²·h. They agree only when r = h for the cone and only for one specific shape; in general they are different. If your shape is a hemisphere, use a sphere calculator and halve the result.

Mistake 6, Treating r = 0 or h = 0 as valid. The formulas degenerate at r = 0 (the "line cone") or h = 0 (the "flat cone"), but they aren't meaningful, every card will read zero. Pick any positive radius and positive height for a real answer. The reverse-mode h = 0 input would also produce a division by zero; the calculator guards against that with explicit h > 0 and r > 0 checks in the inverse formulas.

Frequently Asked Questions

What is the formula for the volume of a cone?

The volume of a right circular cone is V = (1/3)·π·r²·h, where r is the radius of the circular base, h is the perpendicular height from the base to the apex, and π is the circle constant approximately equal to 3.14159265358979. The result is in the same cubic units as r²·h. So a cone with radius 1 and height 1 has a volume of π/3 ≈ 1.04720 cubic units; a cone with radius 0.1 m and height 0.3 m has a volume of about 0.001·π ≈ 3.14 × 10⁻³ m³, or about 3.14 litres.

What is the formula for the slant height of a cone?

The slant height of a right circular cone is l = √(r² + h²), where r is the radius of the base and h is the perpendicular height. The slant is the length of the line from any point on the rim of the base to the apex, measured along the surface of the cone. It is the hypotenuse of the right triangle whose legs are r and h. For the 3-4-5 cone (r = 3, h = 4) the slant is exactly 5; for the 5-12-13 cone (r = 5, h = 12) the slant is exactly 13; for a cone with r = h = 10 the slant is 10·√2 ≈ 14.142.

What is the formula for the surface area of a cone?

The total surface area of a right circular cone is A_total = π·r² + π·r·l = π·r·(r + l), where r is the radius of the base, l is the slant height, π·r² is the area of the base, and π·r·l is the area of the curved lateral surface. If you want only the lateral area (the side, excluding the base), it is A_lateral = π·r·l. For the unit cone (r = h = 1, l = √2), the total area is π + π·√2 ≈ 7.58448 square units.

Who discovered the formulas V = (1/3)πr²h and A_lateral = πrl?

Archimedes of Syracuse proved V = (1/3)·π·r²·h around 225 BCE in his treatise On the Sphere and Cylinder, where he showed that any cone occupies one-third of its circumscribing cylinder. The lateral-area formula π·r·l was known to the ancient Greeks and follows from unrolling the cone's lateral surface into a circular sector. Archimedes' result is the geometric statement that the cone's volume is exactly one-third the volume of the cylinder with the same base radius and the same height.

Can I use the calculator in reverse, from volume and height back to the radius, or from volume and radius back to the height?

Yes. Set the mode to "Reverse: V, h → r" and type a known volume together with a known height; the tool computes r = √(3V / (π·h)) and reports the slant, base area, lateral area, and total area. Set the mode to "Reverse: V, r → h" and type a known volume together with a known radius; the tool computes h = 3V / (π·r²) and reports the same four derived quantities. Both inversions are algebraically exact; no iterative approximation is needed.

Does the calculator work in any unit?

Yes. Pick any length unit for the radius and the height, millimetres, centimetres, metres, kilometres, inches, feet, yards, and the volume, slant, and three areas are reported in the corresponding cubic and square units. The factors (1/3, π) are the same in every system; the only unit conversion is at the display layer. Common pattern: size in metres for engineering work, size in centimetres for schoolbook work, size in inches for US-shop work.

Why does doubling the radius multiply the volume by 4 and the area by 2?

Because the volume formula has r² and the area formulas have r. If r goes from 1 to 2, r² goes from 1 to 4 (so the volume goes up by a factor of 4 for fixed h), and r goes from 1 to 2 (so the lateral area π·r·l goes up by a factor of 2 for fixed l, and the base area π·r² goes up by a factor of 4 for the same reason as the volume). The total area π·r² + π·r·l is a sum of two pieces that scale differently, so its scaling with r alone is not a clean factor; the cleaner statement is to scale r and h together and read off the volume as (1/3)·π·r²·h. Algebraically, that's the difference between "how much stuff fits inside" (r²·h, three-dimensional) and "how much skin covers the outside" (a sum of r² and r·l, which is two-dimensional but not a pure r² scaling because of the slant).

Why is the cone's volume one-third the cylinder's volume?

This is the geometric content of Archimedes' proof. The argument: slice both the cone and the cylinder at the same height h. At height y above the base, the cylinder's cross-section is a disk of area π·r² (constant), while the cone's cross-section is a disk of area π·(r·(h-y)/h)² = π·r²·(h-y)²/h² (linearly shrinking to zero at the apex). The cone's cross-section area is proportional to (h-y)², so integrating from y = 0 to y = h gives one-third of the cylinder's π·r²·h integral. The same integral identity gives the volume of a pyramid as one-third of the prism with the same base and height; the cone is the rotation-symmetric case of the pyramid.

How accurate are the computed values?

The calculator uses the IEEE 754 double-precision value of π (exact to about 15 significant digits). The numerical routines for square root in JavaScript are spec'd to be correctly rounded, so the rounding error in the final answer is at the last digit of an IEEE 754 double, well below any measurement precision available in everyday physics. For schoolbook, engineering, or hobbyist work, treat the results as exact. Round-trip check: if you enter a forward (r, h) → V result into the reverse mode with the same h, you should get back exactly the same r; the calculator's reverse algebra preserves this to all displayed digits.

Also try these free tools:

• Sphere Volume Calculator
• Volume Calculator
• Cylinder Volume Calculator
• Circle Calculator

References

The cone volume and surface area formulas implemented here are the classical Archimedean forms. The primary historical source is Archimedes' On the Sphere and Cylinder (c. 225 BCE), reproduced in T. L. Heath, The Works of Archimedes (Cambridge, 1897, reissued by Dover). Modern treatments of the surface-of-revolution derivation appear in the NIST Handbook of Mathematical Functions (DLMF, Cambridge, 2010); volume and surface area are defined as SI derived quantities under ISO 80000-3:2006 (Quantities and units, Part 3: Space and time). Pythagorean-triangle examples (3-4-5, 5-12-13, 8-15-17) follow directly from the Euclidean definition of the right triangle in Book I of Euclid's Elements (c. 300 BCE).