Cone Volume Calculator
Type the base radius and height of a right circular cone — the calculator returns volume, slant height, lateral and total surface area, all using exact π.
Volume (V = ⅓·π·r²·h)
37.699112 cubic units
- Diameter (d = 2r)
- 6
- Slant height (ℓ = √(r²+h²))
- 5
- Base area (π·r²)
- 28.27
- Lateral surface (π·r·ℓ)
- 47.12
- Total surface (π·r² + π·r·ℓ)
- 75.4
A right circular cone has volume V = ⅓·π·r²·h — one third of the cylinder that would enclose it. The slant height ℓ is the straight-line distance from the apex to the edge of the base, ℓ = √(r²+h²). Lateral surface unrolls to a circular sector of area π·r·ℓ; adding the circular base π·r² gives the total surface.
How to use this calculator
Type the radius r of the circular base and the height h (the perpendicular distance from the centre of the base up to the apex) in any linear unit you like — centimetres, metres, inches or feet — and stay consistent between the two. The calculator returns the volume in cubic units of that unit, alongside the diameter, slant height, base area, lateral (curved) surface and total surface area. Linear outputs use the same unit, areas use squared units, the volume uses cubed units.
How the calculation works
A right circular cone is defined by two numbers: the radius r of its circular base and the perpendicular height h from the centre of the base to the apex. The volume is V = ⅓·π·r²·h — exactly one third of the cylinder of the same radius and height. The slant height ℓ is the straight-line distance from any point on the edge of the base up to the apex; by Pythagoras, ℓ = √(r²+h²). If you unroll the curved side flat, it forms a circular sector of radius ℓ and arc length 2π·r, so its area is π·r·ℓ. Adding the circular base π·r² gives the total surface area π·r·(r+ℓ). π is taken at full machine precision (~16 significant figures), so rounding happens only at display.
Worked example
Take the classic 3-4-5 cone with r = 3 and h = 4. Slant height ℓ = √(3²+4²) = √25 = 5. Base area = π·3² = 9π ≈ 28.274334. Volume = ⅓·9π·4 = 12π ≈ 37.699112 cubic units. Lateral surface = π·3·5 = 15π ≈ 47.123890 square units. Total surface = 9π + 15π = 24π ≈ 75.398224 square units. If your r and h are in centimetres, the volume is in cm³ (millilitres); in metres, m³; in inches, in³ (1 in³ ≈ 16.387 mL).
Frequently asked questions
What is the formula for the volume of a cone?
V = ⅓·π·r²·h, where r is the radius of the circular base and h is the perpendicular height from the base to the apex. The factor of ⅓ comes from integration — a cone has exactly one third the volume of a cylinder with the same base and height, a result Archimedes proved over 2,000 years ago.
What is the difference between height and slant height?
Height (h) is the perpendicular distance from the centre of the base straight up to the apex — the axis of the cone. Slant height (ℓ) is the diagonal distance from the edge of the base to the apex along the curved side. The two are linked by Pythagoras: ℓ = √(r²+h²). Volume uses the perpendicular height; lateral surface uses the slant height. Mixing them up is the most common cone-calculation error.
How do I find the lateral surface area of a cone?
Lateral surface area = π·r·ℓ, where ℓ is the slant height. The intuition: if you cut the cone up the side and unroll the curved face, it lies flat as a sector of a circle of radius ℓ. The sector's arc length equals the base circumference 2π·r, so its area is ½·ℓ·(2π·r) = π·r·ℓ. This calculator computes ℓ from r and h automatically, so you only need to enter the perpendicular height.
Does this work for an oblique cone?
The volume formula V = ⅓·π·r²·h still holds for an oblique cone (where the apex is offset and the axis is tilted) as long as h is the perpendicular height between the base plane and the apex, not the axial length. The surface-area formulas in this calculator assume a right cone, where the apex sits directly above the centre of the base — for an oblique cone the slant height is no longer the same all the way around, so π·r·ℓ does not apply.
How do I convert the volume to litres or gallons?
If your radius and height are in centimetres, the volume comes out in cubic centimetres, and 1 cm³ = 1 millilitre, so divide by 1000 to get litres. In metres, the volume is in m³ and 1 m³ = 1000 litres. For US gallons, 1 US gallon ≈ 3.78541 litres; for UK gallons, ≈ 4.54609 litres. Use the Volume Converter for any other unit pair.
Why is a cone exactly one third of a cylinder?
Cavalieri's principle and integration both show that the volume of a cone of radius r and height h equals ⅓·π·r²·h, while a cylinder of the same dimensions has volume π·r²·h. The ⅓ factor is exact, not an approximation — it comes from integrating π·r(z)² along the axis, where r(z) = r·(1 − z/h) shrinks linearly from r at the base to 0 at the apex. Three identical cones fit perfectly inside one matching cylinder.