A cone has a circular base that narrows evenly to a single apex. This calculator determines how much space a right cone occupies from its base radius and vertical height, a calculation used for piles of material, funnels, hoppers, containers, decorations, and geometry problems. The answer is the internal or solid volume enclosed by the shape, reported in cubic units rather than the square units used for surface measurements.
Quick Answer
The volume of a cone is V = (1/3)πr²h, where r is the base radius and h is the perpendicular height from the base plane to the apex.
Why a Cone Holds One Third of Its Cylinder
A cylinder with the same base and height has volume πr²h. A cone with those dimensions holds exactly one third as much, provided it is a standard cone measured to its vertical height. That factor of one third is not an approximation; it is part of the geometry. It also provides an excellent plausibility check: calculate the matching cylinder mentally and expect the cone to be one third of it.
The radius controls the circular base area, πr². Because the radius is squared, widening a cone has a larger effect than making it a little taller. Doubling r multiplies volume by four, while doubling h only doubles it. Height means the shortest perpendicular distance from the apex down to the plane of the base. It is not the sloping edge length, known as slant height, which is longer and belongs primarily in lateral surface-area calculations.
For a right circular cone, the apex sits above the center of the base. An oblique cone leans, and although some volume relationships can still be developed from its base and vertical height, ordinary school and design problems generally mean the right cone described here. If you are comparing storage shapes, the Cylinder Volume Calculator gives the matching full-height container.
How to Use the Cone Volume Calculator
- Measure the cone's base diameter if the radius is not already known.
- Divide the diameter by two to obtain the radius.
- Measure the vertical height from the base to the tip, not along the sloping side.
- Use the same length unit for radius and height.
- Enter the radius and vertical height.
- Interpret the result in cubic units, such as cubic centimeters or cubic feet.
Filling a Small Conical Hopper
Suppose a hopper has a base radius of 3 feet and a vertical height of 7 feet. Square the radius to get 9. Multiply by the height: 9 × 7 = 63. Include π and then take one third: (1/3) × π × 63 = 21π. Numerically, the hopper volume is 65.97 cubic feet. The matching cylinder would hold 197.92 cubic feet, exactly three times as much.
Notice how the arithmetic can be simplified before reaching for decimals. Keeping 21π until the end avoids repeated rounding and makes the relationship to the cylinder obvious. For manufacturing estimates, round only the final displayed capacity and then add any separate allowance required by the material or filling method.
- A radius of 4 units and a height of 9 units give 150.80 cubic units.
- A radius of 2 units and a height of 5 units give 20.94 cubic units.
- A radius of 6 units and a height of 3 units give 113.10 cubic units.
Slant Height, Diameter, and Capacity Traps
Substituting slant height for vertical height is the classic cone-volume error. The slant runs along the outside surface and is always longer than the perpendicular height in a non-flat cone, so it inflates the result. If only the radius and slant height are supplied, recover the vertical height with the Pythagorean relationship before applying the volume formula, or measure the actual vertical drop.
Diameter and radius mistakes are equally costly. Entering a diameter of 8 as a radius of 8 doubles the radius and multiplies the result by four. Pay attention to whether a funnel specification describes the wide opening, the narrow outlet, or the whole solid. This calculator's cone model uses one circular base and one apex; a truncated cone, or frustum, has two circular ends and needs a different formula.
Volume also is not the same as usable payload in every setting. Granular material may form voids, liquids should not be filled above a safe line, and thin-walled containers have an interior capacity slightly below the volume calculated from exterior dimensions. The formula provides the geometric capacity; application-specific fill factors come afterward. A related round shape with a continuously curved profile can be checked with the Volume of a Sphere Calculator, while other solids and measures appear in the Mathematics Calculators category.
How Resizing a Cone Changes Its Capacity
Scaling makes the squared radius easy to feel. Start with a cone of radius 3 and height 6. Its volume is (1/3)π × 9 × 6 = 18π. Increase only the height to 12 and the volume doubles to 36π. Increase only the original radius to 6 while returning the height to 6, and the volume becomes (1/3)π × 36 × 6 = 72π—four times the initial amount. Increase both dimensions by a factor of two and volume rises eightfold, because the radius contributes a factor of four and height another factor of two.
This sensitivity explains why a small measurement error at a wide base matters more than the same error near a narrow tip. When estimating a pile, measure the base radius at the widest stable footprint and the height at the center. Material angle, settling, and compaction can still make a real pile depart from the ideal cone, but the geometric scaling tells you which measurement deserves the most care.
For a cone-shaped pile measured outdoors, repeat the height and base measurements after the material settles if the estimate will be used for payment or inventory. Freshly poured material can compact, changing height without a matching change in footprint. The formula remains the same, but using settled dimensions gives a more defensible capacity.
Matching the Formula to the Shape in Front of You
Before entering numbers, trace the shape: one circular base, straight tapered sides, and one point. If those features match and the height is measured straight down from that point, the one-third cylinder rule is the right model. A quick radius check and a comparison against the full-cylinder volume will catch most entry errors before the result is used.