MihsabaShare this free calculator
Send this tool to someone who needs the calculation.
How to use
Use the cone calculator to find volume, total surface area, lateral area, base area and slant height from two compatible measurements. Enter radius or diameter with vertical height, or pair the base measurement with slant height. If you know diameter, halve it before using cone formulas. A diameter of 10 cm and perpendicular height of 12 cm give radius 5 cm and slant height 13 cm: volume is 100π cm³ and total surface area is 90π cm². Use lateral area alone when the circular base is excluded.
- Choose whether the base measurement is radius or diameter.
- Choose whether the second measurement is vertical height or slant height.
- Enter positive values in one consistent length unit.
- Review radius, height, slant height, volume, base area, lateral area and total area.
How it is calculated
Example
For radius 3 cm and height 4 cm, the slant height is 5 cm. Volume is 12π ≈ 37.699 cm³, base area is 9π, lateral area is 15π and total surface area is 24π ≈ 75.398 cm².
Important notes
These formulas are for a right circular cone whose apex lies above the center of the base. When slant height is entered, it must be greater than the radius to produce a positive vertical height. Volume is cubic; areas are squared.
Worked examples and interpreting results
A right cone with radius 3 cm and perpendicular height 4 cm has slant height 5 cm. Slant height belongs in the lateral-area formula; perpendicular height belongs in the volume formula.
| Case | Calculation | Result |
|---|---|---|
| Slant height | √(3² + 4²) | 5 cm |
| Volume | π × 3² × 4 ÷ 3 | 37.699112 cm³ |
| Total surface area | π × 3 × (3 + 5) | 75.398224 cm² |
| Volume: diameter 10 cm, height 12 cm | π × (10 ÷ 2)² × 12 ÷ 3 | 314.159265 cm³ |
| Total area: radius 5 cm, slant height 13 cm | π × 5 × (5 + 13) | 282.743339 cm² |
How to check the result
Diameter is twice the radius. Using diameter as radius produces an incorrect volume. These formulas apply to a right circular cone, and total area includes the base.
Cone surface area and volume with solved examples
For a right circular cone, volume equals π times radius squared times perpendicular height divided by three. Multiply π by radius and slant height for lateral area. Add the base area πr² to obtain total surface area, including the circular base.
- Volume for radius 3 and height 4: π × 3² × 4 ÷ 3 = ≈ 37.70
- Lateral area for slant height 5: π × 3 × √(3² + 4²) = ≈ 47.12
- Total area including the base: π × 3 × (3 + √(3² + 4²)) = ≈ 75.40
Calculation caution
Slant height is not the vertical height. Curved surface area excludes the base; total surface area includes the base disk.
Related calculators
Frequently asked questions
How do I calculate cone volume?
For a right circular cone, V = πr²h/3. The calculator applies this after converting your chosen base measurement to radius.
How do I calculate the surface area of a cone?
Total surface area is πr(r+l), where r is radius and l is slant height. The tool also shows lateral and base areas separately.
Is a cone's volume the same as a cylinder's?
No. With the same base and height, the cone has one third of the cylinder's volume.
What is the difference between height and slant height?
Height is perpendicular from the apex to the base center; slant height runs along the surface to the base edge.
Can I enter diameter instead of radius?
Yes. Choose diameter and the calculator converts it to radius internally.
How do I find cone volume from radius and slant height?
First calculate h = √(l² − r²), then V = πr²h/3. Slant height l must exceed radius r. Select the slant-height input option.
What is the volume of a cone with radius 3 and height 4?
Volume equals πr²h/3. Substituting r=3 and perpendicular height h=4 gives V = π × 9 × 4/3 = 12π cubic units, about 37.699. The slant height is not the height used in the volume formula.
What is the difference between lateral and total cone surface area?
Lateral area = πrℓ, where slant height ℓ = √(r²+h²). Total area also includes the circular base: πr(ℓ+r). If r=3 and h=4 then ℓ=5, lateral area=15π and total area=24π square units.