Cone Volume Calculator
Get the volume, slant height and surface areas of a right circular cone from its radius and height.
Formula and sources updated · Data sources · 1View algorithm, constants and default inputs (JSON)
Volume (unit³)
About this calculator
A cone holds exactly one third of the cylinder that would enclose it, so the volume is one third of πr²h. The slant height is the straight-line distance from the rim to the tip — you need it for the curved surface, because the side unrolls into a sector of radius equal to the slant.
Worked example
calculated with the default values below
Base radius 5 · Height 12 Volume 314.159 unit³ · Curved surface area 204.204 unit² · Total surface area 282.743 unit² · Base area 78.540 unit² · Slant height 13.0000 unit
Method and assumptions
V = πr²h/3; slant height = sqrt(r² + h²). Ideal Euclidean shapes; use consistent length units. A diameter is twice the radius.
Data sources
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
V = π × r² × h ÷ 3; l = √(r² + h²); A_l = π × r × l
- r
- base radius
- h
- height
- l
- slant height
- V
- volume
- A_l
- lateral area
Total surface area = lateral area + πr² (base area).
Frequently asked questions
Should I use the slant height or the perpendicular height?
The perpendicular height — the vertical distance from base to apex. Slant height runs along the curved surface and is always longer, so entering it overstates the volume, often by a lot for a wide cone. It is the single most common error in this calculation, and the two values are only equal for a cone of zero radius.
Why is a cone exactly one third of a cylinder?
Because the two have identical base areas and heights, and the cone's volume is one third of the cylinder's at every level of the integration. The same one-third factor holds for any pyramid over any prism with a matching base and height, which is why the two formulas look so similar for shapes that seem unrelated.
Which inputs and limits apply to Cone Volume Calculator?
Use the units shown beside each field. These are form limits, not recommended settings. Base radius: 0.01–100,000; Height: 0.01–100,000 V = πr²h/3; slant height = sqrt(r² + h²). Ideal Euclidean shapes; use consistent length units. A diameter is twice the radius.
What is a complete worked example for Base radius = 5; Height = 12?
Inputs: Base radius = 5; Height = 12. Calculation: V = π × r² × h ÷ 3; l = √(r² + h²); A_l = π × r × l. Results: Volume = 314.159 unit³; Curved surface area = 204.204 unit²; Total surface area = 282.743 unit²; Base area = 78.540 unit²; Slant height = 13.0000 unit.