Cone Volume Calculator
Find the volume of a right circular cone from its radius and height, plus its slant height and surface area.
How it works
Volume = ⅓ · π · r² · h · Slant ℓ = √(r² + h²) · Lateral = π · r · ℓ · Total SA = π · r · (r + ℓ)
A cone fills exactly one third of the cylinder with the same base and height — a fact from calculus that holds for every cone. Its slant height ℓ, the distance from the rim up to the apex, comes from the Pythagorean theorem on the radius and vertical height. The curved (lateral) surface unrolls into a sector of radius ℓ giving area πrℓ, and adding the circular base πr² gives the total surface area.
Worked example
For radius 3 and height 5: volume = ⅓ × π × 3² × 5 = 15π ≈ 47.1239 cubic units. The slant height is √(3² + 5²) = √34 ≈ 5.8310, so the lateral area is π × 3 × 5.8310 ≈ 54.9564 and the total surface area is π × 3 × (3 + 5.8310) ≈ 83.2225.
Sources: Cone — Wikipedia
Frequently asked questions
Why is a cone one third of a cylinder?
Integrating the shrinking circular cross-sections from base to apex gives exactly ⅓πr²h, one third the cylinder πr²h of the same radius and height. Three identical cones fill one cylinder — a result the ancient Greeks proved geometrically.
What is the difference between height and slant height?
The height h is the straight vertical distance from the base to the apex. The slant height ℓ is measured along the sloping surface. They relate by ℓ = √(r² + h²), so the slant is always longer than the vertical height.
Which surface area should I use?
Use the lateral area πrℓ for the curved side alone — say, the paper of an ice-cream cone. Use the total surface area πr(r + ℓ) when the flat circular base is included, such as a solid cone or a closed container.
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This calculator is for educational purposes. Double-check important results before acting on them.