Cone Volume Calculator
A cone is one third of its circumscribing cylinder, a fact the Greeks proved geometrically and every engineer now uses as a sanity check. This calculator computes the volume of a cone from its base radius and vertical height, and also shows the slant height, the lateral area and the total surface area. Good for pile volumes, funnels, party hats and drill-tip shapes: measure the radius, or measure the diameter and halve it, then enter it together with the height.
How to calculate cone volume
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1
Enter the radius
The distance from the centre of the circular base to its edge. If you measured the diameter, divide it by 2 first.
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2
Enter the height
The vertical height from the centre of the base to the apex, measured at a right angle to the base.
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3
Read the results
The volume, slant height, lateral area and total surface area are shown and update live as you type.
Formulas
Volume: V = (1/3) × π × r² × h
Slant height: s = √(r² + h²), the distance from the apex to the rim of the base, measured along the side.
Lateral area (the curved side, base excluded): A = π × r × s
Total surface area (side plus base): A = π × r × (r + s)
Quick sanity check
A cone with radius 3 and height 9 has volume (1/3) × π × 9 × 9 = 27π ≈ 84.82. The enclosing cylinder would be π × 9 × 9 = 81π ≈ 254.47, exactly three times the cone. If your calculator disagrees, you probably fed it diameter where it wanted radius.
Radius vs. diameter: the classic mistake
The formula wants radius, not diameter. If you measure across a circular base with a tape measure, you get the diameter; divide by 2 before plugging in. Doubling this mistake doubles the linear term, which quadruples the volume.
Where cone volume shows up
| Application | What the cone represents |
|---|---|
| Stockpile estimates | Gravel, salt, grain heaps (angle of repose ~30-40°) |
| Funnels and hoppers | Cones with the tip cut off, parallel to the base |
| Traffic cones | Full cone approximation |
| Ice-cream cones | Classic example, rounded tip |
| Drill bits | Tip is a cone, shank is a cylinder |
Stockpile estimation
For a free-standing pile of material with angle of repose θ poured in a circular footprint of radius r, the height is r × tan(θ), giving volume (π × r³ × tan(θ)) / 3. Gravel has θ ≈ 37°, so a pile 8 ft across (r = 4) is about 3 ft tall and contains about 50 cubic feet, a useful check when a truckload arrives.
Frequently Asked Questions
(1/3) × π × 25 × 10 = 250π / 3 ≈ 261.8 cubic units. Cubic inches if you measured in inches, cubic metres if you measured in metres.
Yes, when they share the same base radius and height. This is a classical result: V_cone = V_cylinder / 3. It holds for any pointed solid with a straight taper.
The slant height is the distance from the apex to the rim of the base, measured along the side of the cone. It is not the vertical height: the calculator derives it as s = √(r² + h²) and uses it for the lateral and total surface area.
No. As long as the apex is above the base plane (any tilt), the volume is still (1/3) × base area × perpendicular height. Only the apex-to-centre offset changes the shape, not the volume.
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