Cone Calculator
Calculate volume, surface area, and slant height of a cone. Enter values for instant results with step-by-step formulas.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Volume = (1/3)πr²h | Surface Area = πr² + πrl | l = √(r² + h²)
A cone's volume is exactly one-third that of a cylinder sharing the same base radius and height. Its total surface area combines the flat circular base (πr²) with the curved lateral surface (πrl), where the slant height l is found via the Pythagorean theorem from the radius and vertical height.
Worked Examples
Example 1: Ice cream cone volume
Problem:An ice cream cone has a radius of 3 cm and a height of 10 cm. Find its volume and slant height.
Solution:Volume = (1/3)π(3²)(10) = (1/3)π(9)(10) = 30π ≈ 94.25 cm³. Slant height = √(3² + 10²) = √109 ≈ 10.44 cm.
Result:Volume ≈ 94.25 cm³, Slant height ≈ 10.44 cm
Example 2: Traffic cone surface area
Problem:A traffic cone has a base radius of 18 cm and a slant height of 45 cm. Find its total surface area.
Solution:Surface Area = πr² + πrl = π(18²) + π(18)(45) = 324π + 810π = 1134π ≈ 3562.6 cm².
Result:Surface Area ≈ 3,562.6 cm²
Frequently Asked Questions
Why is a cone's volume exactly one-third of a cylinder's?
A cone and a cylinder that share the same base radius and height have volumes in an exact 1:3 ratio — Archimedes proved this over 2,000 years ago using a method of exhaustion (an early precursor to integral calculus). Intuitively, if you filled a cone-shaped container with water and poured it into a cylinder of the same radius and height, it would take exactly three cone-fuls to fill the cylinder.
How do I calculate a cone's total surface area, and what does each part represent?
Total surface area = πr² + πrl, where πr² is the flat circular base and πrl is the lateral (side) surface area — the curved part you'd get if you unrolled the cone's side into a flat sector. If you only need the lateral surface (like calculating paper needed for a party hat, which has no base), use just πrl.
What is a right circular cone versus an oblique cone?
A right circular cone has its apex directly above the center of its circular base, so the axis is perpendicular to the base — this is the standard cone shape and what the volume and surface area formulas here assume. An oblique cone has its apex offset to one side, but remarkably, Cavalieri's principle shows it has the same volume as a right cone with equal base area and height, even though its surface area formula is more complex.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy