Cone Volume Calculator
V = ⅓ π r² h, with slant height & surface area
How to Use
- 1Enter the radius of the circular base.
- 2Enter the perpendicular height (base to apex, straight up).
- 3Pick your unit: cm, m, inches or feet.
- 4Read the volume plus slant height and surface areas in the tiles.
Cone Formulas
- VolumeV = ⅓ π r² h
- Slant heightℓ = √(r² + h²)
- Lateral surfaceL = π r ℓ
- Total surfaceS = π r (r + ℓ)
Quick Examples
Tap an example to load it above.
Everything about Cone Volume
Why the ⅓ appears, slant height vs height, and worked examples.
A cone's volume is exactly one third of the cylinder that would enclose it: V = ⅓ π r² h. This cone volume calculator also derives the slant height ℓ = √(r² + h²) and both surface areas, showing each substitution step.
For a cone with radius 4 cm and height 9 cm: V = ⅓ × π × 16 × 9 = 48π ≈ 150.80 cm³. The matching cylinder holds 452.39 cm³, three times as much.
Height vs slant height: h is measured straight up the middle; the slant ℓ runs along the outside surface. The formula needs h, so if you only know ℓ, then h = √(ℓ² − r²).
Why one third?
Fill a cone with water and pour it into a cylinder of the same base and height, and it takes exactly three pours. Calculus proves it: integrating the shrinking circular cross-sections from base to apex gives precisely ⅓ of base × height. The same ⅓ appears in pyramid volume for the same reason.
Worked examples
- 1Ice-cream cone
r = 3 cm, h = 12 cm → ⅓ × π × 9 × 12 ≈ 113 cm³, before the scoop on top.
- 2Funnel
r = 6 cm, h = 10 cm → ⅓ × π × 36 × 10 ≈ 377 cm³ ≈ 0.38 litres.
- 3Gravel pile
Loose material settles into a cone. r = 1.5 m, h = 1 m → ⅓ × π × 2.25 × 1 ≈ 2.36 m³.
- 4Party hat (surface, not volume)
Paper needed is the lateral surface πrℓ, so for r = 8 cm, h = 15 cm: ℓ = 17, giving π × 8 × 17 ≈ 427 cm².
Frequently Asked Questions
What is the cone volume formula?
V = ⅓ π r² h, one third of pi, times the radius squared, times the height. It is exactly one third of the volume of the cylinder with the same base and height.
Why is cone volume one third of a cylinder?
Because the circular cross-sections shrink linearly from the base to a point at the apex. Integrating those shrinking circles gives exactly ⅓ of base area × height, and you can verify it by pouring: three cones of water fill the matching cylinder.
What is slant height and how is it different from height?
Height (h) is the straight vertical distance from base to apex; slant height (ℓ) runs along the sloped surface. They relate by Pythagoras: ℓ = √(r² + h²). Volume uses h, lateral surface area uses ℓ.
How do I find cone volume from the diameter?
Divide the diameter by 2 to get the radius, then apply V = ⅓πr²h. Example: d = 8 cm, h = 9 cm → r = 4 → ⅓ × π × 16 × 9 ≈ 150.8 cm³.
What is the surface area of a cone?
Lateral (the slanted surface) is πrℓ; adding the circular base πr² gives the total πr(r + ℓ). For r = 4, h = 9: ℓ ≈ 9.849, lateral ≈ 123.76, total ≈ 174.02 square units.
Is this cone volume calculator free and does it work on mobile?
Yes. Completely free, no sign-up, works in any browser on any device, and every result comes with the substituted formula so you can follow the working line by line.