What is the volume of a cone?
A cone is a 3D shape with a circular base that tapers to a single point (the apex or tip). The most common kind is a "right circular cone" — the apex sits directly above the centre of the base. Ice cream cones, traffic cones, party hats, and the conical lower part of a tornado are all examples.
The volume of a right circular cone is given by:
V = (1/3) × π × r² × h
where r is the radius of the base and h is the perpendicular height (from the centre of the base straight up to the apex). The (1/3) coefficient is the part most people get wrong — it's there because a cone takes up exactly one-third of the volume of the cylinder that fits around it. Drop the 1/3 and your answer is three times too big.
How to use the cone volume calculator
- Enter the radius of the base (r) — for example, 5.
- Enter the perpendicular height (h) — for example, 10. Important: this is straight-up height, not slant height.
- Volume appears instantly: (1/3) × π × 25 × 10 ≈ 261.8.
- Below the volume you also get slant height (the outside-of-cone distance), lateral area (the cone's curved side), and total surface area (curved side + circular base).
- Tap any worked example to load real-world cones — ice cream cone, traffic cone, megaphone — and watch the values update.
Worked examples
Example 1 — Unit cone (r = 1, h = 1)
V = (1/3) × π × 1 × 1 = π/3 ≈ 1.047. The unit cone has slant height √2 ≈ 1.414, lateral area π√2 ≈ 4.443, total surface area π + π√2 ≈ 7.585. It's a tall, narrow cone — taller than it is wide at the base.
Example 2 — Ice cream cone (r ≈ 4 cm, h ≈ 12 cm)
V = (1/3) × π × 16 × 12 ≈ 201 cm³ = 201 mL. About a fifth of a litre — that's how much ice cream actually fits inside a typical cone (excluding the scoop on top, which is its own sphere or hemisphere). Slant height ≈ √(16+144) ≈ 12.6 cm — the length of the wafer cone from base to tip along the outside.
Example 3 — Traffic cone (r ≈ 15 cm, h ≈ 30 cm)
V = (1/3) × π × 225 × 30 ≈ 7069 cm³ ≈ 7 litres. A standard 30 cm orange traffic cone has about 7 litres of internal volume — though most traffic cones are weighted at the base for stability rather than filled solid. Slant height ≈ 33.5 cm.
Example 4 — Megaphone (r = 100 cm, h = 200 cm)
V = (1/3) × π × 10000 × 200 ≈ 2.094 × 10⁶ cm³ ≈ 2094 litres. A massive 1-metre-radius cone, 2 metres tall, would hold about 2 cubic metres of air. Real megaphones are much smaller, but the shape is the same.
Why (1/3)?
The 1/3 coefficient surprises everyone the first time they see it. Why one-third?
Geometrically: imagine a cylinder with the same base radius and same height as your cone. The cone fits perfectly inside the cylinder. The cone's volume is exactly one-third of the cylinder's. You could prove this with calculus (integrate πr(z)² dz from base to tip, where r(z) shrinks linearly from r to 0), or you could prove it physically — fill a hollow cone three times with water and pour it into the matching hollow cylinder. The cylinder fills up exactly.
This 1/3 isn't unique to cones. Any "pyramid" shape — a 3D figure with a flat base that tapers to a single apex — has volume (1/3) × base area × height. So a square pyramid has volume (1/3) × s² × h. A triangular pyramid has volume (1/3) × triangle area × h. The cone formula is just the case where the base is a circle, so base area = πr².
Slant height vs perpendicular height
Two different measurements, often confused:
- Height (h) — the perpendicular distance from the centre of the base straight up to the apex. This is the value the volume formula needs.
- Slant height (ℓ) — the distance from a point on the edge of the base to the apex, measured along the outside of the cone. This shows up in the surface-area formulas.
The two are related by the Pythagorean theorem: ℓ = √(r² + h²). You can think of the slant height as the hypotenuse of a right triangle whose legs are r (the base radius) and h (the perpendicular height).
If a problem gives you slant height instead of height, use h = √(ℓ² − r²) to get the height before plugging into V = (1/3)πr²h.
Surface area: lateral and total
A cone has two distinct surface-area concepts depending on whether you care about the base.
Lateral surface area (no base)
A_lateral = π × r × ℓ
This is the area of the cone's curved side — the part you'd touch if you ran your finger from the base to the apex. The formula is surprisingly clean: it's exactly the same as the area of a sector of a circle (which is what you'd get if you "unrolled" the cone onto a flat surface). Use this when the cone is open at the bottom — a party hat, a megaphone, the cone-shaped paper cup.
Total surface area
A_total = πrℓ + πr² = lateral area + base circle area
Use this when the cone has a base — a traffic cone, an ice cream cone with a wafer disc at the bottom (rare), or a closed conical tank. It's just the lateral area plus the circular base's area (πr²).
Where cones show up in everyday life
- Food and drink — ice cream cones, paper cups, conical filter coffee drippers, party-popper hats. Cones make grippy and pourable shapes.
- Safety equipment — traffic cones, road cones, construction cones. The conical shape is stable (wide base) and visible from all angles.
- Speakers and microphones — speaker cones, megaphones, gramophone horns. The taper amplifies sound by funneling air pressure waves to the smaller end.
- Geology — volcanoes form roughly conical shapes as ash and lava pile up around the central vent. Cinder cones are the most cone-like; stratovolcanoes are more complex.
- Mathematics and physics — light cones (the boundary of cause and effect at relativistic speeds), the apex of a tornado funnel, conical pendulums, conic sections (circle, ellipse, parabola, hyperbola — all from slicing a cone).
- Manufacturing — funnels, conical drill bit tips, conical fasteners (some screw heads), nose cones on rockets and aircraft. The taper helps with insertion, alignment, or aerodynamics.
Common mistakes
- Forgetting the 1/3. πr²h is the cylinder formula. The cone formula has the 1/3 in front. Without it, your answer is 3× too big.
- Confusing slant height with regular height. The volume formula uses perpendicular height (h), not slant height (ℓ). If a problem gives you slant height, convert with h = √(ℓ² − r²) first.
- Using diameter where radius should go. The base radius is half the base diameter. If the problem gives diameter, halve it first.
- Mixing units. Make sure r and h are in the same units. Volume comes out in cubic units of whatever you input (cm³, in³, ft³, etc.).
- Confusing lateral area with total area. Lateral excludes the base; total includes it. Pick the right one for whether your real-world cone is open or closed.
What the calculator gives you, summarized
- Volume — V = (1/3)πr²h, the cone's interior capacity.
- Slant height — ℓ = √(r² + h²), the outside-of-cone distance from base edge to apex.
- Lateral area — πrℓ, the curved surface only.
- Total surface area — πrℓ + πr², lateral plus the circular base.
Two inputs (radius + height), four outputs. The 1/3 coefficient is doing all the work — without it you'd have a cylinder.