How this calculator works
A cone holds exactly one third of the cylinder that fits around it, hence the factor of 1/3 in the volume. The slant height runs from the tip to the edge of the base and is the hypotenuse of the right triangle formed by the radius and the perpendicular height. The lateral surface area is the curved side alone; the total adds the circular base. Negative measurements are rejected, and zero radius or height gives zero volume.
Formula
V = (1/3)πr²h; Slant l = √(r² + h²); Lateral area = πrl; Total area = πr(r + l).
Worked example
A cone with radius 3 and height 4 has slant 5, V = π × 9 × 4/3 = 12π ≈ 37.699, lateral area 15π ≈ 47.124 and total area 24π ≈ 75.398.
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Frequently asked questions
Height or slant height: which do I enter?
Enter the perpendicular height straight down from the tip. The slant height is different, longer, and is calculated for you from the radius and height.
Why one third of a cylinder?
Three identical cones of radius r and height h fill exactly one cylinder of the same radius and height, a classical result often shown by pouring demonstrations.
Does the cone have to stand upright?
No. The volume depends only on the base radius and the perpendicular height, so a slanted cone with the same measurements has the same volume.