A cone has a flat circular base and a curved side that narrows evenly to a single point, the apex. Funnels, traffic cones, ice cream cones, conical roofs and the pile that forms when sand or gravel pours from a chute are all cones. This cone calculator handles the right circular cone, with the tip directly above the center of the base. It finds the volume, capacity, slant height, areas and angles from whichever two measurements you have.
How to use the cone calculator
- Under I know the, choose Radius and height, Radius and slant height, Height and slant height, Volume and radius (find the height), Volume and height (find the radius) or Lateral area and radius (find the height).
- Fill in the fields shown: Base radius (r), Height (h), Slant height (s), Volume (V) in cubic units, or Lateral area in square units (the curved side only).
- Pick the Units.
- The headline is the volume, or the height or radius the calculator solved for. The list gives base radius, height, slant height, volume, capacity in liters and US gallons, lateral area, base area, total surface area, apex angle and side-to-base angle.
Cone formulas
The radius, height and slant height form a right triangle, with the slant height as the hypotenuse, so any two give the third by the Pythagorean theorem. The side-to-base angle is arctan(h ÷ r), and the apex angle is 2 × arctan(r ÷ h).
Worked example: a gravel pile
A delivered pile of gravel is 10 ft across at the bottom and about 4 ft high. How much is there?
- Choose Radius and height, enter r = 5 and h = 4, units feet.
- Volume: ⅓ × π × 5² × 4 ≈ 104.72 ft³. Divide by 27 for about 3.88 cubic yards.
- Slant height: √(5² + 4²) ≈ 6.40 ft, the distance up the side of the pile.
- Side-to-base angle: about 38.7°, a typical slope for loose stone.
Why a cone is one third of a cylinder
Fill a cone with water and pour it into a cylinder with the same base and height, and it takes exactly three loads. The reason is that a cone’s slices shrink in proportion to their distance from the tip, so their areas shrink with the square of that distance, and the average of those squared slices over the height is one third of the base. Archimedes credited Democritus with discovering this and Eudoxus with proving it. Pyramids obey the same rule, which is why the square pyramid calculator also divides by three.
Piles and the angle of repose
When a loose material is poured onto the ground, it forms a cone whose sides settle at a natural slope, its angle of repose. Dry sand typically settles at around 30° to 35°, while coarse, angular stone can stand steeper. Damp or fine material behaves differently, so treat any number as an estimate.
That slope lets you estimate a pile you cannot climb. Height = radius × tan(angle). A road-salt pile 30 ft across that settles at about 32° is roughly 15 × tan 32° ≈ 9.37 ft tall. Entering r = 15 and h = 9.37 gives about 2,208 ft³, or 81.8 cubic yards.
Funnels and paper cones
Unroll a cone’s side and you get a sector of a circle whose radius is the slant height. Its angle is 360° × r ÷ s. For a funnel with r = 3 cm and s = 5 cm (height 4 cm), cut a 216° sector from a circle of radius 5 cm; its area equals the lateral area, 47.12 cm². The sector calculator checks the cut. If the funnel’s tip is cut off, it becomes a conical frustum.
Common mistake: slant height as height
Measuring along the side of a pile or roof gives the slant height, not the height. Using it in the volume formula overestimates: for r = 3 and s = 5, the true volume is 37.70, but treating 5 as the height gives 47.12, 25% too much. Choose Radius and slant height and let the calculator find the real height.
Frequently asked questions
What is the formula for the volume of a cone?
V = 1/3 × π × r² × h, where r is the base radius and h the vertical height. A cone with a 3 cm radius and 4 cm height holds 1/3 × π × 9 × 4 ≈ 37.70 cm³.
What is the difference between height and slant height?
The height runs straight up from the center of the base to the tip. The slant height runs along the sloping side from the base edge to the tip, and it is always longer: s = √(r² + h²). Volume uses the height; the side area uses the slant height.
Why is a cone one third of a cylinder?
A cone's cross-sections shrink steadily from the full base to a point, and adding up those shrinking slices gives exactly one third of the cylinder with the same base and height. You can see it by filling a cone with water or sand: three cone-loads fill the matching cylinder.
How do I estimate the volume of a pile of gravel or sand?
Treat the pile as a cone. Measure across the bottom and halve it for the radius, estimate the height, and choose 'Radius and height'. In feet, divide the cubic feet by 27 to get cubic yards for ordering.
What do the apex angle and side-to-base angle mean?
The apex angle is the full opening angle at the tip, measured across the cone. The side-to-base angle is the slope of the side measured from the ground, the same as a pile's angle of repose. The two always relate as apex angle = 180° − 2 × side-to-base angle.