THE NUMORIX GUIDE
How to use the Cone Calculator
Last reviewed September 14, 2026
What this calculator does
For radius r and vertical height h, slant height = sqrt(r^2+h^2), volume = (1/3)*pi*r^2*h, lateral area = pi*r*s, and total surface area = pi*r*(r+s).
Formula and method
For radius r and vertical height h, slant height = sqrt(r^2+h^2), volume = (1/3)*pi*r^2*h, lateral area = pi*r*s, and total surface area = pi*r*(r+s).
Variables and inputs
Radius defaults to 3 and height to 5. Both are positive lengths; volume is cubic units and areas are square units.
Worked example
For r=3 and h=5: slant height = sqrt(9+25) = sqrt(34) = 5.8310; volume = (1/3)*pi*9*5 = 47.1239; lateral area = pi*3*5.8310 = 54.9554; total area = pi*3*(3+5.8310) = 83.2298 square units.
How to interpret the result
Lateral area excludes the circular base; total area includes it. The height is perpendicular to the base, not the slanted edge.
Common mistakes to avoid
Do not use slant height in the volume formula. Do not omit the base when the requested result is total surface area.
Assumptions and limitations
The formulas describe a right circular cone with a closed circular base. The example and UI assume positive dimensions and ideal geometry.
Practical use and checks
Use the Cone Calculator for a right circular cone such as a hopper, funnel, tent-like solid, or tapering container. Enter radius 3 and vertical height 5 as a check. The slant height should be about 5.8310, volume about 47.1239 cubic units, lateral area about 54.9554 square units, and total surface area about 83.2298 square units. The volume check uses one-third times pi times radius squared times vertical height; substituting the slant height there would be a geometry error. Choose lateral area when only the curved wall needs covering and total area when the circular base is included. Keep the radius distinct from the diameter and use the perpendicular height, not the length of the sloped side, for volume. The model assumes a right circular cone with a complete base and ideal dimensions. It does not include material thickness, an open or cut base, an oblique apex, seams, overlap, or a changing wall profile. For a real container, reserve capacity for headspace and decide whether the physical inside radius or outside radius is relevant. Measurements near zero or a nearly flat cone can magnify rounding, so retain more precision until the final material or capacity decision.