Cone Calculator: Simple Math Tool

The Three Formulas Behind the Results

A cone's slant height, volume, and surface area all derive from just two inputs: radius and height. The slant height comes first because the surface area formula depends on it.

Worked Example: Radius 3, Height 4

Slant height: s = square root of (r squared + h squared) = square root of (9 + 16) = square root of 25 = 5.000.

Volume: V = (1/3) x pi x r squared x h = (1/3) x pi x 9 x 4 = 12 pi = 37.699 cubic units.

Surface area: A = pi x r x (r + s) = pi x 3 x (3 + 5) = 24 pi = 75.398 square units.

The tool also reports two components separately: lateral (side) area = pi x r x s = pi x 3 x 5 = 47.124, and base area = pi x r squared = pi x 9 = 28.274. Adding those two gives back the total surface area (47.124 + 28.274 = 75.398), which is a useful way to sanity-check the output.

Reading the Diagram

The diagram above the inputs labels each variable on a static cone outline: h in red (the vertical height), r in blue (base radius), and s in green dashed (slant height, the hypotenuse of the right triangle formed by r and h). This is the same triangle the Pythagorean formula above solves.

Input Constraints

Both radius and height must be positive numbers; zero or negative values trigger an alert rather than a nonsensical negative volume. There's no upper bound on the values, so very large inputs will produce very large results without warning.