Cone volume and surface area
A cone tapers from a circular base to a point. From the base radius and the vertical height this calculator returns the volume, the slant height, the curved lateral surface, the total surface, and the base area.
The formulas
Volume = one third times pi times radius squared times height — exactly a third of the cylinder that encloses it. The slant height is the square root of (radius squared plus height squared), the straight distance from base edge to apex. Lateral surface = pi times radius times slant height, and total surface adds the base, pi times radius squared.
Why a third of a cylinder
A cone of the same base and height holds one third the volume of the matching cylinder. It is one of the oldest results in solid geometry and a handy mental check.
Related geometry tools
For the straight-walled version, use the cylinder calculator; for the rounded solid, the sphere calculator.
Worked example
A radius of 3 and height of 4 give a slant height of 5 (a 3-4-5 triangle), a volume of about 37.7, a lateral surface of about 47.1, and a total surface of about 75.4.
FAQ
Is slant height the same as height?
No — height is vertical, slant height runs along the sloping surface and is always longer. Surface-area formulas use slant height; volume uses vertical height.
What about a truncated cone?
A frustum — a cone with the tip cut off — uses different formulas with two radii, which this tool does not cover.