Cone Volume Calculator
V = ⅓ π r² h, with slant height, surface area and capacity
Height is measured straight up the middle, from the centre of the base to the apex. If you only have the slant, the calculator shows it back to you so you can check which one your tape actually followed.
Depth is measured from the closed end. Fill a point down cone to half its depth and it holds an eighth of its volume, not half.
Cone formulas
Every figure the panel prints, written out. Volume needs the perpendicular height, the two surface areas need the slant.
- VolumeV = ⅓ π r² h
- Volume from the diameterV = π d² h / 12
- Slant heightℓ = √(r² + h²)
- Lateral surface areaL = π r ℓ
- Total surface areaA = π r (r + ℓ)
- Half angle at the apexθ = arctan(r / h)
- Flat pattern sectorθ = 360° × r / ℓ
- Point down, filled to depth dVd = V (d / h)³
- Point up, filled to depth dVd = V [1 − (1 − d / h)³]
- Free standing pileh = r tan θ
Load a cone into the panel above.
Which measurement do you actually have?
Most cone questions stall because the tape reached something other than r and h. Pick the row that matches what you measured.
| What you have | What to do | Mode |
|---|---|---|
| Radius and height | V = ⅓ π r² h, straight in | Full cone |
| Diameter and height | Switch the base to diameter, or halve it yourself | Full cone |
| Radius and slant height | h = √(ℓ² − r²) first | Full cone |
| Volume and radius | h = 3V / (π r²) | Solve for height |
| Volume and height | r = √(3V / π h) | Solve for radius |
| Base area and height | V = ⅓ × base area × h | Full cone |
| Tape around the base | r = C / 2π, then carry on as normal | Full cone |
| Lateral area and slant | r = L / (π ℓ) | Full cone |
| A depth of liquid inside | Multiply the full volume by (d / h)³ | Partial fill |
A heap of loose material is not a cone of your choosing. It settles at its angle of repose, so the height follows from the base through h = r tan θ. Field values shift with moisture and grading, so treat these as typical rather than exact.
| Material | Angle of repose | Height on a 2 m base radius |
|---|---|---|
| Natural gravel with sand | 25° to 30° | 0.93 m to 1.15 m |
| Wheat | 27° | 1.02 m |
| Dry clay in lumps | 25° to 40° | 0.93 m to 1.68 m |
| Dry sand | 34° | 1.35 m |
| Granite | 35° to 40° | 1.40 m to 1.68 m |
| Snow | 38° | 1.56 m |
| Wet sand | 45° | 2.00 m |
| Crushed stone gravel | 45° | 2.00 m |
The cone, from the one third to the flat pattern
Where the ⅓ comes from, why slant height is not height, what a part filled cone really holds, and how tall a gravel pile stands.
A cone holds exactly one third of the space you might expect from its footprint: V = ⅓ π r² h, where r is the radius of the circular base and h is the perpendicular height measured straight up from the centre of that base to the apex. Give this cone volume calculator those two numbers and it returns the volume of the cone, the slant height, the lateral and total surface area, the half angle at the apex, the flat pattern sector angle, and the capacity in litres with US gallons and Imperial gallons listed separately.
Worked through once, in full. Take a right circular cone of radius 4 cm and perpendicular height 9 cm. The radius squared is 16, so V = ⅓ × π × 16 × 9 = 48π ≈ 150.80 cm³. The slant height is √(16 + 81) ≈ 9.849 cm. The curved surface area is π × 4 × 9.849 ≈ 123.76 cm², and adding the base disc of 50.27 cm² gives a total surface area of about 174.02 cm². Poured out, that cone holds 0.1508 litres, which is 0.0398 US gallons or 0.0332 Imperial gallons.
Where the one third actually comes from
There are three honest answers, and you can take whichever one suits you. The first is a kitchen test: fill a cone with water and empty it into a cylinder with the same base radius and the same height. It takes three cone fills to fill the cylinder, every time, at every size. The second is the shape of the solid itself. Slice the cone horizontally and every slice is a circle, and those circles shrink steadily from the full base radius down to a single point at the apex. Because area depends on the radius squared, the slices lose area faster than they lose height, and the total comes to exactly one third of base area times height. The third answer is Cavalieri's principle: two solids of equal height whose matching horizontal slices have equal area must have the same volume.
That third answer earns its keep, because it settles the leaning cone in one line. Push the apex sideways until the cone tips right over and every horizontal slice keeps exactly the same area it had before, only shifted across. Equal slices, equal volume, so an oblique cone uses the same formula as a right one. The perpendicular height is what counts, not the length of the tilted axis, which is the detail most pages skip.
Height is not slant height, and the picture proves it
The most common wrong answer on a cone comes from measuring up the sloping side rather than up the middle. The perpendicular height h runs from the centre of the base to the apex. The slant height ℓ runs along the outside surface, from the rim to the apex. Those two and the radius form a right angled triangle, so Pythagoras ties them together: ℓ = √(r² + h²), and turned around, h = √(ℓ² - r²). The classic check is a 3, 4, 5 cone: a radius of 3 and a height of 4 give a slant of exactly 5.
Volume uses h. Surface area uses ℓ. That split is why the diagram beside the calculator redraws itself to whatever proportions you type, with r, h and ℓ each labelled where it is actually measured. A tall narrow cone and a wide flat one look nothing alike, and seeing your own numbers drawn tends to catch a swapped measurement faster than any paragraph can.
Which measurement do you actually have?
Very few real cones hand you a clean radius and height. If you have the diameter, switch the base input to diameter and the tool halves it for you, which is the same as using V = π d² h / 12. If you have the slant instead of the height, recover the height with h = √(ℓ² - r²) first. If you already know the volume and need a missing dimension, set the solve for control: h = 3V / (π r²) returns the height, and r = √(3V / π h) returns the radius. If all you can reach is a tape around the base, the circumference gives the radius through r = C / 2π. The reference table above pairs each of these with the mode that handles it.
Lateral surface, total surface, and which one you want
The lateral or curved surface area is the sloping skin on its own, L = π r ℓ. It is the paper in a party hat, the metal in a hopper wall, the fabric in a conical lampshade. The total surface area adds the circular base, giving A = π r (r + ℓ). Ask which surfaces actually exist on the object in front of you. A traffic cone or a funnel has no lid and often no base, so the lateral figure is the one to order material against. A solid cone being painted all over needs the total.
A cone filled part way holds far less than you think
This is the question almost nobody answers, and it catches people out constantly. Stand a cone on its point, the way a hopper, a separating funnel or a cone bottom tank does, and fill it to half its depth. It does not hold half its volume. The wetted part is a smaller cone with the same proportions, so the contents go as the cube of the depth ratio: fraction = (d / h)³. Half the depth is one eighth, or 12.5 percent. A tenth of the depth is a thousandth of the contents. Turn the cone the other way up, flat base down, and the empty part is the small cone instead, so the fraction becomes 1 minus (1 minus d/h)³ and filling to half the depth holds seven eighths. The partial fill tab handles both orientations and prints the result in litres and in both kinds of gallon.
Which brings up capacity. Cubic centimetres tell a cook nothing and cubic inches tell a brewer nothing, so every result here is also given as litres, US gallons and Imperial gallons, each labelled. A US gallon is 3.785 litres and an Imperial gallon is 4.546 litres, so a bare figure marked only as gallons is about 20 percent out for half the people reading it.
Piles of sand, gravel and grain
Tip any loose material onto the ground and it settles into a cone, but the height is not yours to choose. The material picks it, through its angle of repose, and the relationship is h = r tan θ. Dry sand sits at about 34 degrees, wet sand and crushed stone at about 45 degrees, wheat at about 27 degrees, snow at about 38 degrees. On a heap with a base radius of 2 metres, dry sand stands about 1.35 m tall and holds roughly 5.65 m³, while crushed stone on the same footprint reaches the full 2 m and holds about 8.4 m³. Field values shift with moisture and grading, so treat those angles as typical rather than exact. In practice you cannot measure the height of a stockpile anyway, so walk a tape around the bottom, convert with r = C / 2π, and let the angle supply the height.
Cutting the flat pattern
To make a cone out of sheet metal, card or fabric you do not cut a triangle. You cut a sector of a circle whose radius is the slant height, and the angle of that sector is θ = 360° × r / ℓ. For the 3, 4, 5 cone that is 360 × 3 / 5 = 216 degrees, so a little over half a disc of radius 5 rolls up into a cone of base radius 3 and height 4. The material it consumes is the lateral area π r ℓ, which is the reason that number is worth printing at all outside a homework answer.
What changes when you scale a cone
Volume is linear in the height and quadratic in the radius, and that asymmetry surprises people. Double the height and the volume doubles. Double the radius and it quadruples, because the radius is squared. Double both and you get eight times the original, the same cube law that makes a part filled cone behave the way it does. Triple the height and double the diameter and the volume rises by three times four, or twelve. You can test any of these in a few seconds by changing one input above and watching the figure move.
Four mistakes worth checking for
- 1Using the diameter as the radius
It gives four times the true volume. Switch the base input to diameter rather than halving it in your head.
- 2Using the slant height as the height
The slant is always the longer of the two, so the answer comes out too big. Convert first with h = √(ℓ² - r²).
- 3Dropping the one third
That is the cylinder answer, three times too large. If your figure matches the matching cylinder line in the results panel, this is what happened.
- 4Writing a volume in square units
Volume is cubic, surface area is square. Three lengths multiplied give cm³, two give cm².
Where this sits in the curriculum, and what to use next
Cone volume is a grade 8 topic under 8.G.C.9 and returns in high school geometry under HS.G.GMD.A.3, where the derivation and the oblique case matter as much as the arithmetic. Two related jobs live elsewhere so this page can stay about cones. A cone is exactly one third of the cylinder that shares its base and height, and that comparison is printed with every result, but straight sided vessels themselves belong to the cylinder volume calculator. If the top of your cone has been cut off flat it is a frustum rather than a cone, and it has a tool of its own at the frustum calculator. For any other solid, or to compare shapes side by side, start at the volume calculator.
Frequently Asked Questions
What is slant height and why does the volume formula not use it?
Slant height ℓ is the distance along the sloping outside surface, from the rim of the base up to the apex. Perpendicular height h is the straight vertical distance from the centre of the base to the apex. Volume depends on how tall the solid stands, so it uses h, while lateral surface area follows the skin and uses ℓ. They are two sides of the same right angled triangle: ℓ = √(r² + h²), and h = √(ℓ² - r²).
Why is a cone exactly one third of a cylinder?
Because the circular cross sections shrink steadily from the full base radius to a single point, and area depends on the radius squared, so the solid loses bulk faster than it loses height. Adding up those shrinking circles gives precisely one third of base area times height. You can also test it with water: it takes three cone fills to fill the cylinder that shares its base and height, at any size.
What is the difference between lateral and total surface area of a cone?
Lateral, or curved, surface area is the sloping skin alone, π r ℓ. Total surface area adds the circular base, giving π r (r + ℓ). For a cone of radius 4 and height 9 the slant is about 9.849, so the lateral area is about 123.76 square units and the total is about 174.02. Use the lateral figure for open shapes such as funnels, party hats and hopper walls.
How do I find the volume when I only know the slant height?
Recover the perpendicular height first with h = √(ℓ² - r²), then use the ordinary formula, which combines into V = ⅓ π r² √(ℓ² - r²). A base radius of 24 with a slant of 25 gives a height of 7 and a volume of 1344π. If the slant is not longer than the radius, the two measurements cannot belong to the same cone.
Does the formula still work if the cone leans over?
Yes. An oblique cone, one whose apex is not above the centre of its base, has the same volume as the upright cone on the same base with the same perpendicular height. Sliding every horizontal slice sideways changes nothing about its area, and Cavalieri's principle says equal slice areas at equal heights mean equal volume. Measure the perpendicular height, not the length of the tilted axis.
I know the volume, how do I get the radius or the height back?
Rearrange the formula. With the radius known, h = 3V / (π r²). With the height known, r = √(3V / π h). The solve for control on the calculator does either one: pick what you want back, type the volume and the dimension you already have, and the full set of results is rebuilt from it. A volume of 200π with a height of 6, for instance, means a radius of 10.
How do I answer in terms of pi instead of a decimal?
Leave π as a symbol and work the rest: V = ⅓ × r² × h, then write π after it. A radius of 4 and a height of 9 give ⅓ × 16 × 9 = 48, so the answer is 48π cubic units, which most answer keys prefer. The results panel prints that exact coefficient on its own line beside the decimal.
A cone is half full by depth, is it half full by volume?
No, and this is the trap. Standing on its point, the filled part is a smaller cone of the same proportions, so the contents go as the cube of the depth ratio: half the depth cubed is one eighth, or 12.5 percent of the volume. Standing on its base it is the other way round and half the depth holds seven eighths. The partial fill tab computes both orientations for any depth you enter, and tells you the depth that would give you half.
How many litres does my cone hold, and how many gallons?
One litre is 1000 cubic centimetres, so divide a cm³ answer by 1000, and a cubic metre is 1000 litres. For imperial measurements, one cubic foot is 28.317 litres and one cubic inch is 0.016387 litres. Gallons need care: a US gallon is 3.785 litres and an Imperial gallon is 4.546 litres, about 20 percent larger, which is why this page always labels the two separately instead of printing a bare gallon figure.
How tall will a pile of sand or gravel be on a given base?
Loose material settles at its angle of repose, and the height follows from h = r tan θ. Dry sand is around 34 degrees, wet sand and crushed stone around 45 degrees, wheat around 27 degrees. A dry sand heap with a base radius of 2 m therefore stands about 1.35 m and holds roughly 5.65 m³. Measure the circumference with a tape and convert with r = C / 2π, since the height of a stockpile is almost impossible to measure directly.
What flat shape do I cut out to make a cone of this size?
A sector of a circle whose radius equals the slant height ℓ, with a sector angle of 360° × r / ℓ. A cone of base radius 3 and height 4 has a slant of 5, so you cut a 216 degree sector from a disc of radius 5 and roll it up. The material it uses is the lateral surface area, π r ℓ, which the results panel gives you alongside the volume.