mathematics calculator
Cone Surface Area Calculator
Calculate a right circular cone's slant height, lateral area, base area, and total closed surface area.
Cone Surface Area Calculator
Total surface area
Enter the radius and perpendicular height of a right circular cone.
Total surface area includes the circular base; this tool does not model an open, oblique, or truncated cone.
Formula version 1.0.0
How to use this calculator
- Enter positive base radius and perpendicular height in one unit.
- Select Calculate.
- Review slant height √(r²+h²).
- Read lateral, base, and total closed surface areas.
Symbols and units
| Symbol | Meaning | Unit or domain |
|---|---|---|
r | Base radius | length |
h | Perpendicular height | length |
l | Slant height | length |
A | Total closed surface area | square length |
Checked examples
3-4-5 cone
Inputs: Radius 3, height 4
Calculation: l=5; lateral=π×3×5; base=π×3²
Result: Total area 24π
The total includes the circular base.
Check: cone-surface-area-core.test.ts
Unit cone
Inputs: Radius 1, height 1
Calculation: l=√2; A=πr(l+r)
Result: π(1+√2)
Slant height exceeds perpendicular height.
Check: cone-surface-area-core.test.ts
Assumptions, rounding, and limits
- The cone is right and circular.
- Radius and perpendicular height are positive and use one unit.
- Total area includes the base.
- Material thickness and seams are excluded.
Frequently asked questions
Is slant height the same as height?
No. Slant height runs along the surface from rim to apex.
Does total area include the base?
Yes. Lateral area excludes it; total closed area includes it.
Why does area scale by the square?
Scaling every length by k scales every area by k².
Sources and review
- OpenStax: Geometric Formulas — Cone geometry
- OpenStax: Surface Integrals — Surface-area context
- Formula version
- cone.surface-area@1.0.0
- Formula review
- Equatara formula review
- Editorial review
- Equatara editorial review
- Last verified
- 2026-10-25
- Applicability
- Ideal closed right circular cone.
- Next review
- On material formula/source change, or by 2028-10-25.
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Found a problem?
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