Cone Volume Calculator
A cone with a radius of 4 and a height of 9 has a volume of about 150.80, exactly one third of the matching cylinder's 452.39. This cone volume calculator uses V = (1/3) pi r squared h and also reports the slant height, lateral surface area, and total surface area for the same two inputs. Enter the base radius and the vertical height, the distance straight up from the center of the base to the tip, and the tool returns every measurement a right circular cone problem usually asks for in one pass. The one-third relationship with the cylinder that shares the same base and height is the whole trick to remembering the formula, and it holds true whether the cone in question is a paper party hat, a scoop of ice cream, or a gravel pile at a job site. Students checking geometry homework, hobbyists sizing a funnel or mold, and contractors estimating loose material all land on the same formula underneath.
Quick answer
A cone fills exactly one third of the cylinder that shares its base and height.
Volume
150.7964
Slant height
9.8489
Lateral surface area
123.7644
Total surface area
174.0299
What this tells you
- •A cone fills exactly one third of the cylinder that shares its base and height.
- •Volume = (1/3) x pi x radius squared x height, so doubling the radius quadruples the volume while doubling the height only doubles it.
- •The slant height runs along the sloped side, sqrt(r squared + h squared), and it only matters for surface area, not volume.
- •Lateral surface area is pi r times the slant height, and the base adds pi r squared more to reach the total surface area.
- •The radius must be entered as a radius, not a diameter, because the formula squares that value directly.
- •Height means the vertical distance from the base plane to the apex, not the length of the slanted edge.
- •The same one-third rule applies to pyramids too, since any cone or pyramid holds one third of the prism or cylinder built on the same base and height.
How to Use
- 1Enter the base radius of the cone, in any consistent unit (inches, cm, feet, meters).
- 2Enter the vertical height, straight up from base center to tip, in that same unit.
- 3Read the volume, with slant height, lateral surface area, and total surface area listed below it.
- 4If you measured the sloped side instead of the vertical height, convert first: h = sqrt(slant squared - r squared).
- 5Keep units consistent between the two fields. Mixing inches for radius with feet for height produces a volume that is off by a large factor.
How It Works
Formula
V = (1/3) pi r^2 hTake the base area, pi r squared, multiply by the height as if filling a cylinder, then keep one third of that result. For radius 4 and height 9, the base area is 50.27, the matching cylinder would hold 452.39, and the cone holds exactly one third of that, 150.80. The slant height comes from the Pythagorean theorem applied to the radius and the height, here sqrt(16 + 81) = sqrt(97) = 9.85, and it feeds directly into the two surface area figures: lateral surface area is pi times radius times slant height, and total surface area adds the circular base, pi r squared, on top of that. The formula assumes a right circular cone, meaning the apex sits directly above the center of a perfectly round base, which covers the vast majority of real cones people need to size.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Radius 4, height 9
One third of pi x 16 x 9. The slant height is 9.85, and the total surface area comes to 174.03.
Ice cream cone, radius 3 cm, height 12 cm
About 113 ml of ice cream fits inside the cone itself. That does not count the scoop that sits above the rim.
Gravel pile 2 m tall with a 3 m radius
Loose material naturally settles into a cone shape. This is how yards or cubic meters of gravel, sand, and mulch get estimated from a pile's footprint and peak height.
Traffic cone, 12 in across the base, 18 in tall
A standard road cone is mostly hollow plastic. This volume figure describes the space the cone's outer shape encloses, not the material used to build it.
Shop funnel, 10 cm across the top and 20 cm tall
That is just over half a liter of capacity if the funnel were sealed at the bottom. Comparing this figure across funnel sizes helps before pouring liquid or granular material through one.
Grain hopper cone, 8 ft radius and 6 ft deep
That converts to about 14.9 cubic yards. A farm or feed mill uses a figure like this to size the cone-bottomed section under a silo or hopper.
Cone Volumes for Common Dimensions
Volume, slant height, and total surface area by radius and height.
| Radius | Height | Volume | Slant height | Total surface area |
|---|---|---|---|---|
| 2 | 6 | 25.13 | 6.32 | 52.30 |
| 3 | 4 | 37.70 | 5.00 | 75.40 |
| 4 | 9 | 150.80 | 9.85 | 174.03 |
| 5 | 12 | 314.16 | 13.00 | 282.74 |
| 6 | 8 | 301.59 | 10.00 | 301.59 |
Radius 6, height 8 is a coincidence worth noticing: the volume and the total surface area land on nearly the same number, 301.59, purely because of that particular ratio of radius to height. It is not a general rule.
Where Cone Volume Shows Up in Real Life
Cone-shaped volumes come up far more often than the classic geometry-class wedge suggests. Ice cream scoops, paper party hats, road safety cones, kitchen funnels, megaphones, and the cone-bottomed hoppers under grain silos and industrial mixers all reduce to the same right circular cone problem: a round base, a straight vertical height, and a single point at the top. Contractors estimating a pile of sand, gravel, or mulch use the same formula in reverse, treating the loose material's natural resting shape as a cone with a measurable base radius and peak height.
The one-third relationship between a cone and its matching cylinder is not an approximation. It is exact, and it generalizes further than most people expect: any cone or pyramid holds precisely one third of the volume of a prism or cylinder built on the same base shape and the same height, whether that base is a circle, a square, or an irregular polygon. Archimedes proved the circular case more than two thousand years ago, and the same result now falls out of a short calculus integral. A simple physical demonstration still works just as well: fill a cone-shaped cup with water and pour it into a cylinder that shares its base and height, and it takes exactly three pours to fill the cylinder.
Common mistakes
- Dropping the one third. Base area times height gives the surrounding cylinder's volume, and a cone holds only a third of it.
- Entering the slant height as the height. The formula needs the vertical height. A 3-4-5 cone with slant 5 and radius 3 has height 4, not 5.
- Using the diameter as the radius. The radius is squared in the formula, so this single mistake quadruples the calculated volume.
- Mixing units between the two fields, such as a radius in inches paired with a height in feet. Convert both measurements to the same unit before entering them.
- Confusing volume with surface area when the real question is about material. Sizing a paper cup, tent, or hopper's material use calls for surface area, not the volume the cone encloses.
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