Quick answer: the volume of a cone is V = (1/3) π r² h — exactly one-third of the cylinder with the same base and height. Use the vertical height, not the slanted side; if you only have the slant length s, height = √(s² − r²).
The cone volume formula, step by step
where π ≈ 3.14159 · r = base radius · h = vertical height (base to tip)
Worked example — a cone with a 3 cm radius and 5 cm height:
- Square the radius: 3² = 9
- Multiply by π and the height: 9 × 3.14159 × 5 = 141.37
- Divide by 3: 141.37 ÷ 3 = 47.12 cm³
That ÷3 is not an approximation — it's exact, and it applies to every shape that tapers linearly to a point, pyramids included. The intuition: as the cone narrows, its cross-sections shrink with the square of the distance from the tip, and those squares average out to a third of the full base. Formally a cone is a solid of revolution — spin the line y = rx/h about the x-axis and the disk method returns this same ⅓πr²h.
Conical stockpile volumes — quick reference
Dumped gravel, sand and soil settle into cones. At a typical 32° angle of repose, the pile height is roughly 0.31 × its diameter — which lets you estimate a pile from one tape measurement across its base:
| Pile diameter | Typical height | Volume | Cubic yards |
|---|---|---|---|
| 6 ft | 1.9 ft | 17.9 ft³ | 0.66 |
| 8 ft | 2.5 ft | 41.9 ft³ | 1.55 |
| 10 ft | 3.1 ft | 81.2 ft³ | 3.0 |
| 12 ft | 3.7 ft | 139 ft³ | 5.2 |
| 15 ft | 4.7 ft | 277 ft³ | 10.3 |
| 20 ft | 6.2 ft | 649 ft³ | 24.0 |
For material ordering — converting these cubic yards into tons of gravel or sand — the gravel and sand volume calculators add density tables on top of the same math.
Where cone volume shows up in real life
Ice cream, honestly measured
A standard sugar cone about 2 in across and 4.5 in deep holds (1/12)π × 4 × 4.5 ≈ 4.7 in³ — barely a quarter cup before the scoop overhangs. Waffle cones at 3 in × 6 in hold (1/12)π × 9 × 6 ≈ 14.1 in³, three times as much. Shape sells.
Funnels and hoppers
Industrial hoppers drain through a cone. A hopper cone 4 ft across at the top, tapering over 3 ft, holds (1/3)π × 4 × 3 ≈ 12.6 ft³ ≈ 94 gallons of material in the cone section alone. If the taper stops at a chute instead of a point, that's a frustum — use the truncated cone calculator.
Roof turrets and teepees
A conical turret roof 10 ft across and 8 ft tall encloses (1/3)π × 25 × 8 ≈ 209 ft³ of attic space — the number an HVAC installer needs before quoting ventilation.
Common mistakes & pro tips
- Slant height used as height. The sloped side is always longer than the vertical height, so this inflates the result. Convert first: h = √(s² − r²).
- Diameter entered as radius. A 4× error, same as any circular shape. Pile and funnel widths measured across the top are diameters — use Diameter mode.
- Forgetting the ÷3. If your answer exactly triples the calculator's, you computed the cylinder instead.
- Pro tip — measuring a pile's height. Don't climb it. Stand a straight pole vertically beside the pile, sight across the peak, and mark the pole. Or use the 0.31 × diameter estimate from the table above.
- Pro tip — cone + cylinder combos. Silo bottoms and funnels sit under cylindrical sections. Compute each part with its own calculator and add — see the cylinder calculator for the straight section.
How to use this calculator
- Measure the base. Measure the radius or diameter of the circular base. Pick the matching mode in the calculator.
- Measure the vertical height. Use the perpendicular height from base to tip — not the slanted side. If you only have the slant length s, the height is √(s² − r²).
- Let it apply V = (1/3)πr²h. Square the radius, multiply by π and the height, then divide by 3. The calculator shows every substitution.
- Convert the result. Switch the output between cubic units, liters and gallons as needed.
Frequently asked questions
Why is a cone's volume one-third of a cylinder's?
A cone tapers linearly from a full circle to a point, and integrating that shrinking cross-section gives exactly one-third of the cylinder with the same base and height. You can prove it in the kitchen: a conical cup fills its matching cylindrical container in exactly three pours.
How do I find cone volume with the diameter?
Halve the diameter first, or use V = (π/12)d²h. A cone 6 inches across and 8 inches tall: V = (π/12) × 36 × 8 = 75.4 in³. The calculator's Diameter mode does this automatically.
What's the difference between height and slant height?
Height is the straight vertical drop from tip to base center; slant height runs along the sloped surface. Only the vertical height belongs in the volume formula. Convert with h = √(s² − r²): a cone with a 5 cm slant and 3 cm radius has h = √(25 − 9) = 4 cm.
How much gravel is in a cone-shaped pile?
Dumped material settles into a cone at its angle of repose — about 30–37° for gravel and sand. Measure the pile's diameter and height, then apply V = (1/3)πr²h. A pile 12 ft across and 3.5 ft high holds (1/3)π × 36 × 3.5 ≈ 132 ft³ ≈ 4.9 cubic yards.
What is the volume of a cone with radius 3 and height 5?
V = (1/3) × π × 3² × 5 = (1/3) × π × 45 = 47.12 cubic units — exactly one-third of the 141.37 the matching cylinder holds.