What is the cone volume?
Type the base radius and the height to get the volume of the cone, or type the volume and one measurement to get the other.
- Volume
- 25.13 in³
A cone with radius 2 in and height 6 in has a volume of 25.13 in³.
Volume: 25.13 in³. A cone with radius 2 in and height 6 in has a volume of 25.13 in³.
How to calculate
Finds the volume of a cone from its base radius and height (V = πr²h ÷ 3), or the radius or height from the volume.
Example with the default inputs (Radius 2 in, Height 6 in): A cone with radius 2 in and height 6 in has a volume of 25.13 in³.
Formula: V = πr²h ÷ 3: one third of the base area πr² times the height h.
- The cone has a circular base. The height is the vertical distance from the base to the tip, so the formula holds for a slanted (oblique) cone too.
- π is the full-precision value 3.141592653589793….
- The radius and height are in the length units shown next to them, and the volume is in the volume unit shown next to it.
- Each value must be greater than 0.
Worked examples
Each example is checked against the calculator on every build.
- Radius 78.74 in, Height 236.2 in gives Volume 1,533,694 in³.Source: OpenStax Prealgebra 2e, Example 9.55 (25.12 with π ≈ 3.14); hand calculation in content.mdx: π × 2² × 6 ÷ 3 = 8π = 25.1327
- Radius 98.43 in, Height 315 in gives Volume 3,195,196 in³.Source: OpenStax Prealgebra 2e, Example 9.56, diameter 5 (52.33 with π ≈ 3.14); hand calculation in content.mdx: π × 2.5² × 8 ÷ 3 = 52.3599
- Volume 6,102,374 in³, Height 236.2 in gives Radius 157.1 in.Source: hand calculation in content.mdx: r = √(3 × 100 ÷ (π × 6)) = √(50 ÷ π) = 3.9894
- Volume 3,051,187 in³, Radius 78.74 in gives Height 469.9 in.Source: hand calculation in content.mdx: h = 3 × 50 ÷ (π × 2²) = 11.9366
How it works
A cone has a circular base of radius r and a tip at height h above the centre of the base. Its volume V is one third of the base area times the height:
V = πr²h ÷ 3
Type any two of the three values and the calculator finds the third:
- volume from radius and height: V = πr²h ÷ 3
- radius from volume and height: r = √(3V ÷ (πh))
- height from volume and radius: h = 3V ÷ (πr²)
Assumptions
- The base is a circle. h is the vertical height, not the slant height, so the same formula works for a slanted (oblique) cone.
- π is the full-precision value 3.141592653589793…. Textbooks that use 3.14 get answers about 0.05% smaller.
- Each value is greater than 0.
- Each box has its own unit. Lengths are converted through metres and volumes through cubic metres (1 in = 0.0254 m exactly). In one length unit, the volume is in that unit cubed.
Worked examples by hand
Radius 2, height 6 (OpenStax Example 9.55). V = π × 2² × 6 ÷ 3 = 8π = 25.1327. The textbook rounds π to 3.14 and gets 25.12.
Diameter 5, height 8 (OpenStax Example 9.56). The radius is 5 ÷ 2 = 2.5, so V = π × 2.5² × 8 ÷ 3 = 50π ÷ 3 = 52.3599. With π ≈ 3.14 the textbook gets 52.33.
Volume 100, height 6. r = √(3 × 100 ÷ (π × 6)) = √(50 ÷ π) = √15.9155 = 3.9894.
Volume 50, radius 2. h = 3 × 50 ÷ (π × 2²) = 150 ÷ 12.5664 = 11.9366.
Other questions people ask
What is the formula for the volume of a cone?
V = πr²h ÷ 3, or one third of πr²h. The base is a circle of area πr², and h is the height from the base to the tip. A cone with radius 2 in and height 6 in holds π × 4 × 6 ÷ 3 = 25.13 in³.
Why is a cone one third of a cylinder?
A cone and a cylinder with the same base and the same height differ in how fast their cross-sections shrink. Adding up the shrinking circles (with calculus, or by filling a cone three times to fill the cylinder) gives exactly one third.
Do I use the height or the slant height?
The height: the straight-up distance from the centre of the base to the tip. The slant height runs along the side and is longer. If you only know the slant height s, the height is √(s² − r²).
How much does an ice cream cone hold?
A cone 5 cm across (radius 2.5 cm) and 10 cm deep holds π × 2.5² × 10 ÷ 3 = 65.4 cm³, about 65 mL, if it is filled level with the rim.
How do I find the height of a cone from its volume?
Multiply the volume by 3 and divide by the base area: h = 3V ÷ (πr²). A 50 in³ cone with radius 2 in is 150 ÷ 12.566 = 11.94 in tall.
I know the diameter, not the radius. What do I type?
Halve the diameter and type that as the radius. A cone 5 in across has a radius of 2.5 in.