Geometry
How To Calculate The Volume Of A Cone
A cone holds exactly one third of the cylinder that would fit around it, which is where the one third in the formula comes from and why it surprises people.
Quick Answer
volume = pi x radius^2 x height / 3
- V
- Volume inside the cone, in cubic units
- r
- Radius of the circular base
- h
- Perpendicular height from base to apex
Square the radius, multiply by pi, multiply by the height, then divide by three. A cone with a radius of three and a height of seven has a volume of about 65.97 cubic units, exactly one third of the cylinder around it.
What Is The Volume Of A Cone?
The volume of a cone is the space inside it, given by one third of pi times the radius squared times the height. A cone of radius three and height seven holds about 65.97 cubic units of anything you pour into it.
The one third is the interesting part. A cone and a cylinder with the same base and the same height are related by exactly that factor, so three cones fill the cylinder. This was known to the ancient Greeks and is a satisfying demonstration to do with water.
The height in the formula is the perpendicular height, measured straight up from the centre of the base to the apex. It is not the slant height, which runs up the curved surface from the edge of the base to the tip and is always longer.
The radius is half the diameter of the base circle. If you have measured across the base, halve it before squaring. Squaring the diameter instead of the radius is the single most common error and makes the answer four times too large.
Because the radius is squared, the volume grows quickly with width but only linearly with height. Doubling the radius multiplies the volume by four, while doubling the height only doubles it. Wide cones hold far more than tall thin ones of the same height.
The slant height comes from Pythagoras. The radius, the perpendicular height and the slant height form a right triangle, so the slant height is the square root of radius squared plus height squared. For a radius of three and a height of seven that is about 7.62.
Surface area is a separate calculation and is not needed for volume. The curved part has an area of pi times radius times slant height, and the flat circular base adds pi times radius squared. Mixing the two formulas up is easy because both start with pi times radius.
Units matter in the usual way. A radius in centimetres and a height in centimetres give a volume in cubic centimetres, which divides by a thousand to become litres. A radius in metres and a height in centimetres must be converted first.
Cones appear constantly in real objects: ice cream cones, funnels, traffic cones, conical piles of sand or grain, and the roof of a round tower. In each case the volume calculation answers a practical question about capacity or material.
A truncated cone, called a frustum, has the top cut off. Its volume is not simply a scaled cone volume; it uses a slightly longer formula involving both radii. Assuming the simple formula on a frustum underestimates the volume badly.
Finally, the answer is always in cubic units. Volume is three dimensional, so a radius and height in metres give cubic metres. Writing square metres for a volume is the same class of error as writing metres for an area, one dimension short.
Formula
V = pi x r^2 x h / 3
The cylinder formula divided by three. Radius and height must be in the same unit.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| V | Volume | cubic metres | Space enclosed by the cone. |
| r | Radius | metres | Half the diameter of the circular base. |
| h | Perpendicular height | metres | Base centre to apex, measured straight up. |
l = sqrt(r^2 + h^2)
Pythagoras on the right triangle formed by the radius, the height and the sloping edge.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| l | Slant height | metres | Distance up the curved surface from base edge to apex. |
| r | Radius | metres | Radius of the base. |
| h | Perpendicular height | metres | Vertical height. |
A = pi x r x l + pi x r^2
The curved surface plus the circular base. Not needed for volume, but they are often asked for together.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Surface area | square metres | Total outside area of the cone. |
| r | Radius | metres | Radius of the base. |
| l | Slant height | metres | Sloping edge length. |
How To Calculate The Volume Of A Cone
- 1
Find the radius of the base
Halve the diameter if that is what you measured. The formula squares the radius, so using the diameter by mistake inflates the answer by a factor of four.
- 2
Measure the perpendicular height
This is the straight up distance from the base centre to the apex, not the distance along the slope. If you only have the slant height, recover the perpendicular height with Pythagoras.
- 3
Square the radius and multiply by pi
A radius of three squares to nine, and nine times pi is about 28.27. This is the area of the base circle, which is also the cross section of the matching cylinder.
- 4
Multiply by the height and divide by three
28.27 times seven is about 197.92, and a third of that is 65.97. The division by three is what turns the cylinder into the cone that fits inside it.
- 5
Attach cubic units
A radius and height in centimetres give cubic centimetres, which divide by a thousand for litres. A radius in metres and a height in centimetres have to be converted to match first.
Examples
Example 1: A cone with radius three and height seven
- Radius
- 3
- Height
- 7
| Step | Calculation | Result |
|---|---|---|
| Base area | pi x 3^2 | 28.2743 |
| Slant height | sqrt(9 + 49) | 7.6158 |
| Volume | 28.2743 x 7 / 3 | 65.9734 |
Result: A cone of radius three and height seven holds 65.9734 cubic units, exactly one third of the cylinder with the same base and height.
Example 2: A sand pile five metres across
- Radius
- 2.5
- Height
- 2
| Step | Calculation | Result |
|---|---|---|
| Base area | pi x 2.5^2 | 19.635 |
| Slant height | sqrt(6.25 + 4) | 3.2016 |
| Volume | 19.635 x 2 / 3 | 13.09 |
Result: A sand pile five metres across and two metres high holds about 13.09 cubic metres, roughly sixteen tonnes of dry sand.
Example 3: A conical funnel with radius four and height nine
- Radius
- 4
- Height
- 9
| Step | Calculation | Result |
|---|---|---|
| Base area | pi x 4^2 | 50.2655 |
| Slant height | sqrt(16 + 81) | 9.8489 |
| Volume | 50.2655 x 9 / 3 | 150.7964 |
Result: A funnel of radius four and height nine holds 150.7964 cubic centimetres, which is about 0.15 litres of liquid.
Calculator
Volume
65.9734
- Base area
- 28.2743
- Slant height
- 7.6158
- Curved surface area
- 71.777
- Total surface area
- 100.0513
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Volume Of A Cone calculator page.
Common Mistakes
Squaring the diameter instead of the radius
The formula needs the radius, which is half the diameter. Squaring the full width multiplies the volume by four, and the answer still looks like a plausible number.
Using the slant height as the height
The slant height runs up the curved surface and is always longer than the perpendicular height. Substituting it overstates the volume, and it is the natural mistake when the sloping edge is the labelled one.
Forgetting the division by three
A cone is not a cylinder. Omitting the one third gives exactly three times the correct volume, which is the most common arithmetic slip in this calculation.
Mixing units between radius and height
A radius in centimetres with a height in metres produces a meaningless result. Convert both to the same unit before squaring anything.
Giving the answer in square units
Volume is three dimensional, so the unit is cubed. Square centimetres describe the base, not the space inside.
Applying the formula to a frustum
A cone with the top cut off is a frustum and needs its own formula using both radii. Using the plain cone formula on it underestimates the volume substantially.
Confusing volume with surface area
Both formulas start with pi and the radius, but one answers how much fits inside and the other how much material covers the outside. They are not interchangeable.
FAQ
What is the formula for the volume of a cone?
One third of pi times the radius squared times the height. Square the radius, multiply by pi and by the height, then divide by three.
Why is there a one third in the formula?
Because a cone fills exactly one third of the cylinder that shares its base and height. Three identical cones would fill that cylinder completely.
What is the difference between slant height and perpendicular height?
The perpendicular height goes straight up from the base centre to the apex. The slant height follows the curved surface from the edge of the base to the apex, and is always the longer of the two.
How do you find the radius if you know the diameter?
Divide the diameter by two. A base twenty centimetres across has a radius of ten, and it is the ten that gets squared.
Does doubling the radius double the volume?
No, it quadruples it. The radius is squared in the formula, so volume grows with the square of the radius but only linearly with the height.
How do you convert the answer to litres?
If the radius and height were in centimetres, the volume comes out in cubic centimetres. Divide by a thousand to get litres.
References
- [1]Wikipedia, Cone — https://en.wikipedia.org/wiki/Cone
- [2]Wikipedia, Volume — https://en.wikipedia.org/wiki/Volume
- [3]Math is Fun, Cone volume — https://www.mathsisfun.com/geometry/cone.html
- [4]Wikipedia, Frustum — https://en.wikipedia.org/wiki/Frustum