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Free online cone calculator to quickly find the volume, surface area, lateral surface area, and base area of a cone.
Formulas
Slant Height: l = √(r² + h²)
Base Area: A_base = π·r²
Lateral Surface Area: A_lat = π·r·l
Total Surface Area: A = π·r·(r + l)
Volume: V = (1/3)·π·r²·h
Please enter positive real numbers. Both radius (r) and height (h) must be greater than 0.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
Enter the base radius and perpendicular height of a right circular cone. The calculator derives slant height, base area, lateral area, total surface area, volume, and the full and half apex angles. A right circular cone has a circular base and an apex directly above the base center; radius and vertical height form the legs of a right triangle inside the cone.
The slant height is l = √(r² + h²). The base area is πr², lateral area is πrl, and total surface area includes the base: πr² + πrl. Volume is ⅓πr²h. For the apex angle, the half-angle is atan(r/h) converted to degrees, and the full angle is twice that value.
Choose one length unit for both inputs. The length result uses that unit, area results use its square, and volume uses its cube. For example, centimeters produce cm, cm², and cm³. The unit selector labels the results; it does not convert a radius entered in one unit to a height entered in another.
The volume uses perpendicular height, while the curved side area uses slant height. Total area adds the flat circular base to the lateral surface. OpenStax’s calculus text gives the same right-cone surface-area relationship and explicitly adds base area when reporting the total.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Enter the distance from the center of the circular base to its rim, not the full diameter. If you measured diameter, divide it by two first.
Use the straight distance from the base plane to the apex. Do not substitute the slanted side length.
Select mm, cm, m, inches, or feet. Choose 2, 4, 6, or 8 decimal places; displayed values trim unnecessary trailing zeros and may use scientific notation for very small or large results.
Read volume, slant height, base area, lateral area, total surface area, and apex angles. Copy the volume if that is the value you need, and keep the unit with it.
For radius 5 cm and perpendicular height 12 cm, slant height is √(25 + 144) = 13 cm. Volume is 100π cm³, about 314.1593 cm³; base area is 25π cm², lateral area is 65π cm², and total surface area is 90π cm². The full apex angle is about 45.2397° and the half angle about 22.6199°. Changing precision changes only the displayed rounding.
Use cases
See how the tool fits into real work and everyday tasks.
A student can enter radius and perpendicular height, then compare the displayed slant-height and area formulas with a worked solution.
A planner can estimate the ideal geometric volume of a right-cone-shaped pile from consistent measurements, then account separately for irregular surfaces or fill conditions.
A designer can compare a calculated slant length or surface area with a right-cone drawing, keeping the selected unit and rounded display precision visible in notes.
Q&A
Find concise answers to common questions and confusing cases.
The input is radius. Divide the full base diameter by two before entering it.
No. The height field is the perpendicular distance from base to apex. Slant height is calculated from radius and perpendicular height.
No. A frustum has a second circular base and needs additional dimensions and formulas. This calculator models a complete right circular cone.
Total surface area includes both the curved side and one circular base. The base area is added to lateral area.
Notes
Review scope, result limitations, and important precautions before use.
The formulas assume a complete right circular cone with positive radius and height. They do not model an oblique cone, a frustum, an irregular vessel, wall thickness, or measurement uncertainty. Results are rounded for display according to the selected precision; use the underlying measurements and an appropriate engineering method for consequential design or capacity decisions.
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