How to use the cone calculator
- Measure the base radius from the centre of the circular base to its edge. If you measured the complete diameter, divide it by 2 before entering the radius.
- Measure the perpendicular height from the base centre to the apex. Do not use the outside sloping edge as height; that edge is the slant height calculated by this page.
- Choose one length unit for both dimensions. Slant height, square-area, and cubic-volume results all follow that unit.
- Select Calculate or press Enter. Copy results if useful. Editing a value or input unit immediately clears the old result and copy state. Clear removes inputs and results while retaining the selected units.
Cone formulas and result meanings
Slant height: apex to base edge
For a right circular cone, the radius and perpendicular height form the legs of a right triangle. The slant height is s = √(r² + h²). It is a linear length along the cone’s side, not the vertical height.
Volume: one third of the matching cylinder
Volume is V = (1/3)πr²h. A cone with the same base radius and perpendicular height as a cylinder occupies one third of that cylinder’s volume. The answer uses cubic units and assumes a complete solid cone.
Lateral surface area: curved side only
The lateral area is L = πrs = πr√(r² + h²). If the curved surface were unrolled, it would form a circular sector whose radius is the slant height. This result excludes the circular base.
Total surface area: curved side plus one base
The total area is A = πrs + πr² = πr(s + r). It includes the complete curved side and exactly one circular base. Unlike a cylinder, a cone has only one base to add.
Units and rounding
The calculator converts both input lengths to metres, calculates with unrounded values, then converts to the selected output unit. Length, square area, and cubic volume use different conversion powers; rounding affects only the visible answer.
Worked examples
Cone with radius 3 cm and height 4 cm
The slant height is √(3² + 4²) = 5 cm. Volume = (1/3)π × 3² × 4 = 12π ≈ 37.7 cm³. Lateral area = π × 3 × 5 = 15π ≈ 47.12 cm², and total area = 15π + 9π = 24π ≈ 75.4 cm².
Radius 5 cm and height 12 cm
The slant height is 13 cm. Volume = 100π ≈ 314.16 cm³, lateral area = 65π ≈ 204.2 cm², and total surface area = 90π ≈ 282.74 cm².
When these results are useful
- Use slant height when a geometry exercise or idealised cover follows the cone’s outside edge. It is always longer than either positive radius or perpendicular height.
- Use volume for a complete solid cone or ideal internal space. Use lateral area for the curved wrapper only, and total area when one circular base is also present. Add seams, overlap, thickness, or waste separately.
- The diagram shows a right cone whose apex is directly above the base centre. It explains the measurements but is not drawn to scale.
Input limits and common errors
- Enter positive finite decimal values. Do not enter unit symbols, fractions, formulas, or thousands separators in the fields. Use a period for the decimal point and the menus for units.
- This page covers a complete right circular cone only. It does not calculate a frustum, an oblique cone, sheet cutting patterns, material weight, density, or purchasing quantity.
- The total-area result includes one base. If your real object is open at the base, use the lateral area instead. Physical capacity may differ because of wall thickness, a rounded tip, or incomplete filling.
Frequently asked questions
What is the difference between height and slant height?
Height is perpendicular from the base centre to the apex. Slant height runs along the outside from the apex to the base edge and equals √(r² + h²).
Does total surface area include the circular base?
Yes. It includes the curved side and one complete circular base. Lateral area excludes the base.
Can radius and height use different units?
Use one common unit, as in the sphere and cuboid calculators. Convert either measurement first if they were recorded in different units.
Why did the result disappear after editing?
The old result no longer matches the current dimensions, so it is removed and copying is disabled until you calculate again.
Can this calculate a truncated or tilted cone?
No. A frustum has two circular radii, and an oblique cone has different geometry. This page is limited to a complete right circular cone.