Cone Volume Calculator
A cone's volume is (1/3) π r² h: a third of the area of its circular base, π times the radius squared, times the height straight up from the centre of that base to the tip. Type the radius and the height and the answer appears beside a labelled drawing, with the formula above it and the working, your numbers put in, as the first row below. The Measure switch takes a diameter instead when that is what you measured, and the height is the vertical one, not the slanted side. This page opens the calculator on the cone whatever shape you used last, takes metres, centimetres, feet or inches, and reads the answer out in litres, US gallons and UK gallons as well as the cube of the unit you typed in.
Cone · (1/3) π r² h314.159 L · 82.9921 US gal · 69.1054 UK gal
The height is straight up from the centre of the base to the tip, not along the slanted side. A third of the cylinder around it.
The formula is on the stage, and the working is the first row. Every answer here is one textbook formula with your numbers put in, printed so you can check it: a cylinder is π r² h, a sphere is (4/3) π r³, a pool is length times width times the average of its two depths. The horizontal tank uses the exact circular-segment formula for a partial fill, so half way up is exactly half and a quarter of the way up is less than a quarter, because the bottom of a circle is narrow. Conversions use exact definitions: a litre is 1,000 cm³, a US gallon is 231 cubic inches and a UK gallon is 4.54609 litres.
Common questions
- What is the formula for the volume of a cone?
- (1/3) π r² h: the area of the round base, π r², times the height, divided by three. The formula line sits beside the drawing and the first row underneath, The working, repeats it with your numbers in, for example (1/3) × π × 0.5² × 1.2 = 0.314159265359. That third is the whole point of a cone: a cylinder of the same radius of 0.5 m and the same height of 1.2 m holds 0.942478 m³, exactly three times as much, so a cone fills a third of the tin it stands in.
- Do I type the radius or the diameter?
- Either, as long as you tell the tool which. The field is a radius by default and the Measure switch beside it turns it into a diameter, which is the full width across the base and the easier one to measure on something already built. The diameter is halved before it goes into the formula: the formula line then reads (1/3) π r² h, r = d/2 and the working opens with the halving, for instance r = 20 / 2 = 10. A party hat 20 cm across and 30 cm tall is (1/3) × π × 10² × 30 = 3,141.59 cm³, which the litres row gives as 3.14159 L. Putting a diameter into the radius field by mistake gives four times too much, because the radius is squared.
- I measured the sloping side. What do I type as the height?
- Not the sloping side. The height field is the perpendicular height, straight up from the centre of the base to the tip, and the dashed line down the middle of the drawing is the one it labels. The tool does not turn a slant height into a height for you, so work it out first with Pythagoras: h is the square root of the slant squared minus the radius squared. A cone with a 5 cm radius and a 13 cm slant side has a height of 12 cm, because 13² − 5² = 144, and typing 5 and 12 in centimetres gives (1/3) × π × 5² × 12 = 314.159 cm³, or 0.314159 litres.
- How do I get litres or gallons out of a cone?
- They are already under the answer. Whatever unit you measure in, the rows list litres, US gallons and UK gallons, plus three cubic units for the scale you are working at. A paper cone of radius 3 cm and height 12 cm is 113.097 cm³, which the litres row reads as 0.113097 L. A conical hopper measured in feet, radius 2 ft and height 3 ft, is 12.5664 ft³, and the rows give 355.84 litres and 94.003 US gallons. The conversions use exact definitions: a litre is 1,000 cm³, a US gallon is 231 cubic inches and a UK gallon is 4.54609 litres.
- My cone is cut off at the top. Can it do a truncated cone?
- There is no frustum on the shape row, so run the cone twice and subtract: the whole cone down to the imaginary tip, minus the small cone that was cut off. Extend the sloping sides until they meet to get the missing height. A cup 8 cm across the top, 5 cm across the bottom and 10 cm deep has a radius growing from 2.5 cm to 4 cm over those 10 cm, that is 0.15 cm per cm, so the tip sits 2.5 / 0.15 = 16.667 cm below the small end and the full cone is 26.667 cm tall. Radius 4 with height 26.6667 gives 446.805 cm³, radius 2.5 with height 16.6667 gives 109.083 cm³, and the difference, about 337.7 cm³, is the cup: near enough 0.34 litres.
- How big is a conical pile of gravel, sand or mulch?
- A heap tipped off a truck settles into something close to a cone, so measure it across the base for the diameter and up to the peak for the height. With Measure set to Diameter, a pile 3 m across and 1.2 m high is (1/3) × π × 1.5² × 1.2 = 2.82743 m³, and the rows read 2,827.43 litres, 99.8499 cubic feet and 3.69814 cubic yards, which is how aggregate is usually sold. The arithmetic is exact for the numbers you type; a real heap is never a perfect cone, so read it as a close estimate of the load rather than a measurement of it.
- Which units can I measure the cone in?
- Metres, centimetres, feet or inches, picked on the Unit row, and the radius and the height are both typed in that same unit. The volume comes back in the cube of it, m³ from metres and in³ from inches, with litres and both gallons underneath every time. Switching the unit changes how your numbers are read rather than converting them, so 0.5 typed in metres becomes half a foot when you press ft: retype the sides in the unit you picked.
Volumes are computed from the formulas shown, exact for the dimensions you type; a horizontal tank's partial fill is the exact circular-segment result. A US gallon is 231 cubic inches and a UK gallon 4.54609 litres.