Cone Surface Area Calculator
The surface area of a cone from its radius and height — including the slant height it needs along the way.
A cone's sloping surface is a sector of a circle whose radius is the slant height, l = √(r² + h²) — the distance up the slope, not the vertical height.
How the cone surface area calculator works
A cone's sloping surface is a sector of a circle whose radius is the slant height, l = √(r² + h²) — the distance up the slope, not the vertical height. Its area is πrl. Add the circular base of πr² and the total is πr(r + l). The classic 3-4-5 relationship shows up here: a cone of radius 3 and height 4 has a slant height of exactly 5.
Formula: cube 6a²; box 2(lw+lh+wh); sphere 4πr²; cylinder 2πr(r+h); cone πr(r+l); pyramid a² + 2a·s
Worked examples
| Inputs | Total surface area | Note |
|---|---|---|
| Radius 3, height 4 | 75.3982 | slant height 5, total 24π ≈ 75.4 |
| An ice-cream cone, radius 2.5, height 12 | 115.9063 | tall and narrow |
| A wide shallow cone | 642.1512 | the base is most of the area |
FAQFrequently asked questions
What is the slant height of a cone?
The distance from the rim up to the tip along the surface: √(r² + h²). It is always longer than the vertical height.
What is the surface area formula?
πr(r + l) in total, where l is the slant height. The sloping part alone is πrl.
Do I need the base?
Not for an open cone like a party hat or a filter — use the "curved or side area" row instead.
Why is the volume only a third?
A cone holds exactly one third of the cylinder with the same base and height — a result that goes back to Eudoxus and Archimedes.
What if I know the slant height instead of the vertical one?
The vertical height is √(l² − r²); enter that.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.