Cone Volume Calculator
Calculate cone volume from radius and height or slant height. See liters, gallons, step-by-step math, and equivalent cylinder comparisons.
The area of the flat circular base.
A cone holds exactly one-third the volume of a cylinder with identical base radius and vertical height.
The volume of a cone is exactly one-third of a cylinder with the same base and height: V = (1/3) × π × r² × h. For example, a cone with a 4-inch radius and 9-inch height has a volume of 150.80 cubic inches (0.65 US gallons).
Cone Volume Calculator Formula
| Variable | Definition | Standard Units |
|---|---|---|
| V | Internal Volume of the cone | in³, ft³, cm³, or m³ |
| r | Radius of the circular bottom base | in, ft, cm, or m |
| h | Perpendicular vertical height from base to apex | in, ft, cm, or m |
| l | Slant height along the outer sloped side | in, ft, cm, or m |
Why a Cone Holds Exactly One-Third of a Cylinder
If you create a hollow cone and a hollow cylinder with identical circular bases and heights, it takes exactly three full cones of water to fill the cylinder. Mathematically, integrating circular cross-sections that taper linearly from radius r down to 0 at the apex yields a factor of 1/3.
Calculating Cone Volume from Slant Height
In many physical situations—such as measuring a conical roof, funnel, or gravel pile—measuring the vertical height inside the solid object is impossible. You can only measure the sloped outer surface (slant height l). By the Pythagorean theorem, l² = r² + h², which means vertical height is h = √(l² - r²).
Estimating Conical Piles of Sand, Gravel, and Mulch
Bulk materials dumped from conveyor belts naturally form conical stockpiles governed by their angle of repose (typically 30° to 37° for dry aggregate). Measuring the base diameter across the ground and the apex peak height gives an accurate volume estimate in cubic yards.
Common Calculation Mistakes to Avoid
- Confusing slant height along the slope with vertical perpendicular height
- Forgetting the 1/3 divisor and calculating the full cylinder volume
- Entering base diameter when the formula expects radius
- Assuming the cone must be pointed upwards; orientation has zero effect on volume
Frequently Asked Questions
Why is the cone volume formula multiplied by 1/3?▼
How do you find the volume of a cone using slant height?▼
How do you convert cone volume to gallons?▼
What is the slant height of a cone?▼
What is a truncated cone called?▼
Authoritative Sources & Reference Standards
- Guide for the Use of the International System of Units (SI), NIST Special Publication 811 — National Institute of Standards and Technology (NIST). Verified 2026-10-01.
Official SI unit definitions, conversion factors (Appendix B), and precision standards for cubic meters, liters, and US customary units.
- Common Core State Standards for Mathematics: High School Geometry (Geometric Measurement and Dimension HSG.GMD.A.1–3) — National Governors Association Center for Best Practices & Council of Chief State School Officers. Verified 2026-10-01.
Curricular standards for volume formulas of cylinders, pyramids, cones, and spheres, including Cavalieri’s principle.