What the one-third in the cone formula means
A cone holds exactly one third of the cylinder that just contains it. Stand a cone inside a cylinder of the same base radius and the same height, and the empty space around the cone is twice the cone's own volume. That single fact is the whole formula: the cylinder is πr²h, so the cone is πr²h/3.
The reason is that a cone's cross-section shrinks as you go up. At a height y above the base of a cone of total height h, the radius has fallen to r(1 − y/h), so the cross-sectional area has fallen to πr²(1 − y/h)² — the square is what makes the shrinkage so aggressive. Averaging that area over the full height gives exactly one third of the base area, which is why the factor is 1/3 and not, say, 1/2. Euclid proved the result in Book XII of the Elements using the method of exhaustion, centuries before calculus made it a one-line integral.
The same one-third factor governs every pyramid, whatever the shape of its base, and for the same reason. That is why the pyramid volume calculator uses base area × height ÷ 3, and why a cone is best thought of as a pyramid whose base happens to be a circle.
Vertical height and slant height are different numbers
The single most expensive mistake in cone work is substituting the slant height for the vertical height. The vertical height h runs straight up from the centre of the base to the apex; the slant height l runs along the sloping surface from a point on the rim to the apex. They form a right triangle with the radius, so l² = r² + h², which is the Pythagorean theorem applied to that triangle.
Slant height is always the larger of the two, and the gap widens as the cone gets squatter. For the default cone here, r = 3 ft and h = 5 ft, the slant height is √34 = 5.831 ft, only 17% longer than the height. For a wide, shallow pile with r = 10 ft and h = 3 ft, the slant is √109 = 10.44 ft, nearly three and a half times the height. Use the slant in a volume formula in that second case and you overstate the volume by 248%.
Which one you measure depends on what you can reach. On a stockpile you can usually pace the slant with a tape but not the vertical height, so measure the slant and the base radius and let the calculator recover h. On a manufactured hopper the drawing gives you the height and the cone angle. The two surface areas need the slant, not the height: the lateral area of a cone is πrl, which is what you use to work out how much sheet metal a cone takes.
Worked example: a 24-foot-diameter sand stockpile
A conical sand pile measures 24 feet across the base and 7 feet from the ground to the peak. Find its volume in cubic yards, the way a materials estimator would.
- Halve the diameter. r = 24 ÷ 2 = 12 ft.
- Square it. r² = 144 ft².
- Find the base area. π × 144 = 452.389 ft².
- Multiply by the height. 452.389 × 7 = 3,166.73 ft³. That is the enclosing cylinder.
- Divide by three. 3,166.73 ÷ 3 = 1,055.58 ft³.
- Convert to cubic yards. 1,055.58 ÷ 27 = 39.10 yd³.
The slant height, if you needed the tarpaulin to cover it, is √(12² + 7²) = √193 = 13.892 ft, giving a lateral area of π × 12 × 13.892 = 523.68 ft².
Sanity-check the shape while you are there. The side of the pile rises at arctan(h/r) = arctan(7/12) = 30.26° above horizontal, which is the angle the material is standing at. Dry sand rests at a broadly similar angle, so the measurement is plausible; a pile computing to a far steeper angle suggests the height was mismeasured or the base was not circular.
Reading the partial-fill table
A point-up cone fills unevenly, and the partial-fill table is the part of this calculator that professionals use most. Because the widest cross-section sits at the bottom, the first half of the height holds far more than half the volume: filling to 50% of the height captures 1 − (1 − 0.5)³ = 87.5% of the total. Filling to 20% of the height already holds 48.8% of the volume.
Turn that round and the consequence for measurement is stark. If you are estimating a stockpile from a height reading, an error of 5% in the height near the top costs you almost nothing, while the same 5% error near the base is expensive. It also means a nearly full cone looks nearly empty from the side: at 80% of the volume the material only reaches 41.5% of the height.
For a hopper, which is a cone standing point-down, the arithmetic inverts: the volume filled to depth d from the apex is simply V(d/h)³, so half the depth holds only 12.5% of the contents. This calculator's table is built for the point-up case. If your cone stands on its point, read the table's share of full column as the share still to be filled above that level.
Volume of common cone sizes
| Base diameter | Height | Volume (ft³) | Cubic yards | US gallons |
|---|---|---|---|---|
| 2 ft | 2 ft | 2.094 | 0.078 | 15.67 |
| 4 ft | 3 ft | 12.566 | 0.465 | 94.00 |
| 6 ft | 4 ft | 37.699 | 1.396 | 282.01 |
| 10 ft | 5 ft | 130.900 | 4.848 | 979.20 |
| 16 ft | 6 ft | 402.124 | 14.893 | 3,008.10 |
| 24 ft | 7 ft | 1,055.58 | 39.10 | 7,896.25 |
| 30 ft | 9 ft | 2,120.58 | 78.54 | 15,863.00 |
| 40 ft | 12 ft | 5,026.55 | 186.17 | 37,601.19 |
Cubic yards are the trade unit for aggregate and topsoil; gallons are the trade unit for liquids in conical tank bottoms.
Assumptions and limits of this calculator
- The base is a true circle. A stockpile dumped from a fixed point is close to circular; one built by a loader working from one side is not, and its volume can be several per cent off.
- The surface is straight from rim to apex. Real piles slump into a slightly concave profile, which makes the true volume a little smaller than the cone formula predicts.
- Volume is not weight. Multiply by the bulk density of the material to get tonnage; loose aggregate and compacted aggregate have very different densities, and neither equals the solid density of the rock.
- The cone is complete. A truncated cone with a flat top needs the frustum formula instead; use the cone frustum volume calculator for that shape.
- Capacity conversions assume the US gallon of 231 cubic inches. The imperial gallon is 20% larger, so a UK figure will not match.
Key terms
- Slant height
- The distance from a point on the base rim to the apex, measured along the sloping surface. It equals √(r² + h²) for a right circular cone.
- Lateral surface area
- The area of the curved side only, πrl, excluding the circular base. This is the sheet-metal area of a cone.
- Angle of repose
- The steepest angle at which a granular material stands unaided. It sets the shape a free-standing conical pile can actually take.
Related shapes and when to use them instead
If the top of your cone is cut off flat, you have a frustum, and its volume is (πh/3)(R² + Rr + r²) rather than the simple cone formula. Bins, silo transitions and most manufactured hoppers are frustums, so check for a flat top before assuming a point.
If the shape is a cylinder with a cone on top — a grain silo, for instance — compute the two separately and add. The cylinder volume calculator handles the barrel; this page handles the cap. For a rounded dome instead of a cone, the spherical cap volume calculator is the right tool.
For the surface area alone — a paint, wrap or sheet-metal estimate — the cone surface area calculator works directly from the radius and slant height. And when you need the plan area of the ground a pile occupies, that is the base area reported here, or the circle area calculator from the diameter alone.