Volume of a cone

Cone formulas

• Volume: V = ⅓ × π × r² × h
• Slant height: s = √(r² + h²)
• Lateral area: π × r × s
• Total surface area: π × r × (r + s)

A cone holds exactly one third of a cylinder with the same base and height, a fact Democritus knew and Eudoxus proved.

Worked examples

A cone with r = 3 cm and h = 4 cm: V = ⅓ × π × 9 × 4 = 12π ≈ 37.7 cm³, slant height 5 cm, side area 15π ≈ 47.1 cm².

A conical pile of gravel 2 m high and 6 m across (r = 3 m) holds ⅓ × π × 9 × 2 ≈ 18.8 m³, useful when ordering trucks.

Ice-cream cones and funnels

The side area tells you how much paper or metal you need to make a cone: unrolled, the side is a sector of a circle with radius equal to the slant height.

The page multiplies the base area πr² by the height and divides by three for the volume. It finds the slant height with the Pythagorean theorem (the height and radius are the legs of a right triangle), then the side area as π × radius × slant height, and the total surface by adding the round base. Results carry your unit, cubed or squared. Everything runs in your browser.