A 3–4–5 cone cross section
Radius 3 and perpendicular height 4 give slant height 5. Volume is 12π, lateral area is 15π, and total area including the base is 24π.
Cone Calculator inputs
Changing the unit relabels the numbers. Convert measurements to one common unit before entering them.
Perpendicular height h runs from apex to base center. Slant height ℓ runs along the side and must exceed radius r. Positive measurements must be from 0.000000001 to 1,000,000,000; derived defining lengths must also remain in this range.
Calculated measurements
Volume
37.69911184 units³
| Measurement | Value |
|---|---|
| Volume | 37.69911184 units³ |
| Total surface (with base) | 75.39822369 units² |
| Lateral surface (no base) | 47.1238898 units² |
| Base area | 28.27433388 units² |
| Perpendicular height | 4 units |
| Slant height | 5 units |
Formula: ℓ = √(r² + h²); V = πr²h/3; lateral = πrℓ
A full right circular cone with its apex above the center of the base. Truncated and oblique cones are excluded.
Results use full floating-point precision internally and display up to 10 significant digits. Angles are in degrees.
Calculations run locally in your browser. Measurement values are not sent to a calculation service. The downloadable text contains your selected inputs and computed results.
How to use the Cone Calculator
- 01
Choose whether the second measurement is perpendicular height, slant height, or volume.
- 02
Enter the base radius and the requested positive measurement.
- 03
Check the labeled apex-to-center height and apex-to-edge slant height in the diagram.
- 04
Read volume and surface results, then copy or download them.
Examples
Recover a vertical height
Radius 5 with slant height 13 gives perpendicular height 12. The slant height must be strictly greater than the radius.
Work back from volume
A cone with radius 2 and volume 8π has perpendicular height 6. The equivalent cylinder with the same base and height has three times the volume.
Vertical height and slant height
A right cone places its apex directly above the base center. Radius r, perpendicular height h, and slant height ℓ form a right triangle through the axis, with ℓ² = r² + h². A tape measurement down the side is therefore the slant height, which is longer than the perpendicular height.
Compute volume from a circular base
Volume is πr²h/3, one third of a cylinder with the same circular base and perpendicular height. When volume is known, height is 3V/(πr²). Slant height cannot replace h in that equation: doing so overstates volume because the sloping side is longer.
Curved area and the base disk
Lateral surface is πrℓ. The closed total adds the base disk πr². Use the lateral output for the ideal curved skin of an open cone; use total surface when the base is present. Neither result includes seams, tabs, cut losses, rim thickness, or the reverse side of a sheet.
A full cone is different from a frustum
Removing the pointed top creates a truncated cone, or frustum, which has two radii and another surface formula. Moving the apex away from the center creates an oblique cone. Those shapes cannot be recovered by substituting one convenient height into this right-cone interface.
What to know
- Full right circular cones only; no frustums, oblique cones, or thickness model.
- Derived perpendicular and slant heights must remain within 10⁻⁹–10⁹ units.
Common questions
Can slant height equal radius?
No. That would leave perpendicular height zero. A nondegenerate cone requires slant height to exceed radius.
Does total surface include the base?
Yes. Total surface equals curved lateral surface plus one circular base. The two parts are also listed separately.
Can I use a cone-shaped container’s outside dimensions?
Those describe the outside geometry. Use inside dimensions to approximate capacity and account separately for thickness, fittings, and usable fill level.
Are units converted automatically?
No. Convert all lengths to the selected unit before entering them, and enter volume in the corresponding cubic unit.