Total Surface Area
A = πrl + πr²
Add the curved surface area to the area of the circular base.
Cone Surface Area Calculator
Calculate the total surface area, curved surface area, circular base area, slant height, and volume of a right circular cone. Enter a radius or diameter and provide either the vertical height or the slant height. Results include formulas, unit conversions, and step-by-step calculations.
Calculate Cone Surface Area
Enter positive dimensions using the same length unit. Choose the type of height you know and select your preferred output units.
Total surface area
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Enter valid dimensions to calculate.
Total surface area
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Curved surface area
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Base area
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Slant height
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Vertical height
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Base circumference
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Volume
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Total surface area includes the curved surface and one circular base. Volume uses vertical height, not slant height.
Cone Surface Area Formulas
A right circular cone has a circular base and a curved surface that narrows to a single tip. The radius describes the base, the vertical height is perpendicular to the base, and the slant height runs along the curved side from the edge of the base to the tip.
Curved Surface Area
Ac = πrl
This measures only the curved outer surface, excluding the base.
Base Area
Ab = πr²
The cone has one circular base. Its area is calculated using the radius.
Slant Height
l = √(r² + h²)
Use this when the radius and vertical height are known.
Vertical Height
h = √(l² − r²)
Use this when the radius and slant height are known. The slant height must be greater than the radius for a cone with positive height.
Volume
V = ⅓πr²h
The volume is one-third of the volume of a cylinder with the same base and vertical height.
Symbols: \(r\) = base radius, \(d\) = base diameter,
\(h\) = vertical height, \(l\) = slant height, and \(\pi\) is approximately
3.141592653589793. The formulas assume a right circular cone.
Worked Examples
These examples show how the cone formulas work with common dimensions. All lengths within each example use the same unit.
Example 1: Radius 3 m and vertical height 4 m
First calculate the slant height using the Pythagorean theorem.
l = √(3² + 4²) = √25 = 5 m
Then calculate the curved surface area.
Ac = π × 3 × 5 = 15π ≈ 47.1239 m²
Calculate the circular base area.
Ab = π × 3² = 9π ≈ 28.2743 m²
Add both areas for the total surface area.
A = 15π + 9π = 24π ≈ 75.3982 m²
Finally, calculate the volume using vertical height.
V = ⅓ × π × 3² × 4 = 12π ≈ 37.6991 m³
Example 2: Radius 5 cm and slant height 13 cm
Since slant height is given, find the vertical height first.
h = √(13² − 5²) = √144 = 12 cm
Curved surface area:
Ac = π × 5 × 13 = 65π ≈ 204.2035 cm²
Base area:
Ab = π × 5² = 25π ≈ 78.5398 cm²
Total surface area:
A = 65π + 25π = 90π ≈ 282.7433 cm²
Volume:
V = ⅓ × π × 5² × 12 = 100π ≈ 314.1593 cm³
Quick reference table
| Radius | Vertical height | Slant height | Total surface area |
|---|---|---|---|
| 3 m | 4 m | 5 m | 24π ≈ 75.3982 m² |
| 5 cm | 12 cm | 13 cm | 90π ≈ 282.7433 cm² |
| 2 in | 3 in | √13 ≈ 3.6056 in | 2π(√13 + 2) ≈ 35.2036 in² |
How to Use the Cone Calculator
- Enter the base measurement. Choose radius if you know the distance from the center to the edge, or diameter if you know the full distance across the circular base.
- Choose the height you know. Select vertical height for the perpendicular distance to the tip, or slant height for the distance along the cone's side.
- Select the input unit. Use the same unit for the base measurement and the height.
- Choose output units and calculate. Select the area and volume units, then press Calculate to see the results and the highlighted value.
- Review, copy, or share your results. The calculator shows total area, curved area, base area, slant height, vertical height, circumference, and volume.
Length and Area Unit Conversions
Convert all input dimensions to a common unit before applying the formulas. This calculator converts inputs internally and lets you select separate output units for area and volume.
| Length unit | Equivalent in meters | Area conversion to m² |
|---|---|---|
| Millimeter (mm) | 0.001 m | 1 mm² = 0.000001 m² |
| Centimeter (cm) | 0.01 m | 1 cm² = 0.0001 m² |
| Meter (m) | 1 m | 1 m² = 1 m² |
| Kilometer (km) | 1,000 m | 1 km² = 1,000,000 m² |
| Inch (in) | 0.0254 m | 1 in² = 0.00064516 m² |
| Foot (ft) | 0.3048 m | 1 ft² = 0.09290304 m² |
| Yard (yd) | 0.9144 m | 1 yd² = 0.83612736 m² |
Area units scale by the square of the length conversion factor. Volume units scale by the cube of the factor. For example, because 1 ft = 0.3048 m, 1 ft³ = 0.3048³ m³ = 0.028316846592 m³.
Frequently Asked Questions
1. What is the formula for the total surface area of a cone?
The formula is A = πrl + πr². Multiply pi by the radius and slant height to find the curved surface area, then add the circular base area, πr².
2. How do I calculate the curved surface area?
Use Ac = πrl. You need the base radius and slant height. If you know only the vertical height, calculate the slant height first with \(l=\sqrt{r^2+h^2}\).
3. What is the difference between vertical height and slant height?
Vertical height is measured perpendicularly from the center of the base to the cone's tip. Slant height follows the angled side from the edge of the base to the tip. They are not interchangeable.
4. Can I use diameter instead of radius?
Yes. Radius is half the diameter: r = d ÷ 2. The calculator accepts either measurement and converts diameter to radius automatically.
5. Does total surface area include the base?
Yes. For a solid cone, total surface area includes the curved surface and one circular base. Curved surface area excludes the base.
6. How is cone volume calculated?
Use V = ⅓πr²h. The height in this formula is the vertical height. If you know slant height instead, the calculator derives the vertical height from the radius and slant height.
7. Can the slant height be smaller than the radius?
Not for a right circular cone with positive vertical height. The slant height must be greater than the radius. If they are equal, the vertical height is zero and the shape is degenerate rather than a normal cone.
8. What units should I use?
Use consistent length units for the radius or diameter and the height. Surface area is expressed in square units, while volume is expressed in cubic units. You can select output units separately in this calculator.
9. Is this calculator suitable for every cone?
The formulas here are for a right circular cone, whose tip lies directly above the center of its circular base. An oblique cone may require different geometric information to determine its lateral surface area.