Cone Volume Calculator — V = (1/3)πr²h
Enter the base radius and perpendicular height of a right circular cone to instantly get its volume, base area, slant height, lateral surface area and total surface area.
units
units
V = (1/3)·π·r²·h
37.7 u³
- 1
Base area (π·r²)
π × 3² = 28.2743 - 2
Volume ((1÷3)·π·r²·h)
(1÷3) × 28.2743 × 4 = 37.6991A cone holds exactly one-third the volume of the cylinder with the same base and height.
How does this calculator work?
A cone holds exactly one-third the volume of a cylinder with the same base radius r and height h: V = (1/3)·π·r²·h. Enter radius and perpendicular height to get volume (units³), base area π·r² (units²) and slant height l = √(r² + h²).
Formula
How this is calculated
A right circular cone is a 3-D solid with a flat circular base of radius r and a single apex located directly above the centre at height h. Its volume is exactly one-third the volume of the cylinder that encloses it (same r and h): V = (1/3)·π·r²·h. This factor of 1/3 can be proven by Cavalieri's principle or by integration — it is exact, not an approximation.
The slant height l = √(r² + h²) is the straight-line distance from the apex to any point on the base rim, derived from Pythagoras applied to the right triangle formed by r, h and l. The lateral (curved) surface area is π·r·l — imagine unrolling the cone into a flat sector. Adding the circular base π·r² gives the total surface area π·r·(l + r).
All formulas apply to a right circular cone with a perfectly flat base and a single apex on the central axis. For a frustum (truncated cone) or for determining the volume of an irregular cone, different approaches are needed. Results are in cubic and square units matching whatever unit you input.
Frequently asked questions
A cone and a cylinder with the same base radius and height satisfy Cavalieri's principle: every horizontal cross-section of the cone is a circle with area that scales as (distance from apex)², and integrating this over the height gives exactly (1/3)πr²h. Equivalently, you can fill an open cone exactly three times from the same cylinder.
The formula V = (1/3)·π·r²·h applies to any cone (or pyramid) with a circular base, as long as h is the perpendicular height from the apex to the base plane — not the slant height. Oblique cones (apex off-centre) still satisfy this formula; only the slant height and surface area formulas change.
Rearrange Pythagoras: h = √(l² − r²). Then substitute into V = (1/3)·π·r²·h. Both l and h must produce a positive result — if l < r the geometry is impossible.
TG we-Calculate Editorial Team. (2026). Cone Volume Calculator — V = (1/3)πr²h [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cone-volume-calculator
TG we-Calculate Editorial Team. "Cone Volume Calculator — V = (1/3)πr²h." TG we-Calculate. 2026. https://we-calculate.com/calculator/cone-volume-calculator.
TG we-Calculate Editorial Team, "Cone Volume Calculator — V = (1/3)πr²h," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cone-volume-calculator
@misc{wecalculate_cone_volume_calculator, title = {Cone Volume Calculator — V = (1/3)πr²h}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cone-volume-calculator}}, year = {2026}, note = {TG we-Calculate} }
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