How this calculator works
A right circular cone has a circular base and an apex directly above the centre. The radius, vertical height and slant height form a right triangle, so any two of them determine the third through the Pythagorean theorem. Volume is one third of the matching cylinder. The base area is πr², the curved lateral area is πr times the slant height, and the total surface area adds the base to the lateral area.
Formula
Slant height l = √(r² + h²); volume V = πr²h/3; base area A_B = πr²; lateral area A_L = πrl; total area A = πr(r + l).
Worked example
A cone of radius 3 m and height 4 m has a slant height of 5 m. Its base area is 9π ≈ 28.274 m², its lateral area is 15π ≈ 47.124 m², its total surface area is 24π ≈ 75.398 m² and its volume is 12π ≈ 37.699 m³.
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Frequently asked questions
What is the slant height?
It is the straight-line distance from the rim of the base to the apex, measured along the curved surface. It is always longer than the vertical height unless the cone has zero radius.
Why is the volume one third of a cylinder?
A cone with the same base and height fills exactly one third of the cylinder that encloses it. The same one-third factor appears in every pyramid and cone volume formula.
Does the calculator need both radius and height?
No. Enter the radius with the slant height, or the height with the slant height, and the missing one is reconstructed from the right triangle. If the slant height is shorter than the other given side, no such cone exists and the input is rejected.