Cone Volume Calculator
The cone volume formula, explained and computed: enter radius and height to get the volume, exact and decimal, with substitution steps.
How to use this calculator
Enter the radius of the base and the vertical height (not the slant height), then press Calculate. The calculator applies V = (1/3)πr²h and shows the exact and decimal results with each step.
How it works
A cone holds exactly one-third of the cylinder with the same base and height, pour water from the cone into the cylinder three times to fill it. So the formula is the cylinder formula with a 1/3 factor: V = (1/3)πr²h.
Worked example
Example: r = 3, h = 6
Step 1, Substitute: V = (1/3) × π × 3² × 6.
Step 2, Simplify: V = (1/3) × π × 9 × 6 = 18π.
Step 3, Decimal: V ≈ 56.549 cubic units.
Common mistakes
- Dropping the 1/3: the single most common cone error, without it you have computed a cylinder.
- Using slant height as h: the formula needs the vertical height. Slant height ℓ converts via h = √(ℓ² − r²).
- Diameter for radius: halve it first, as with cylinders.
When to use it
Cones shape funnels, traffic cones, volcano profiles, ice-cream cones, and conical tanks. Volume tells you capacity, how much a hopper holds, or the concrete in a conical pile.
Frequently asked questions
V = (1/3)πr²h: one-third times pi times radius squared times vertical height.
A cone occupies exactly one-third of the volume of a cylinder with the same base and height. Calculus (or a water-pouring experiment) proves the factor.
Vertical height. If you only know the slant height ℓ and radius r, find h = √(ℓ² − r²) first with the Pythagorean theorem.
V_cone = V_cylinder ÷ 3 for matching base and height. Three cones of water fill one cylinder.