Key takeaways
- The volume of a cone is V = (1/3) × π × r² × h.
- A cone always holds exactly one-third as much as a cylinder with the same base and height.
- Volume uses the vertical height, never the slanted side.
- Forgetting the 1/3 is the most common mistake with this formula.
In this guide
What is the volume of a cone?
A cone is a 3D shape with a flat circular base that narrows smoothly to a single point (the apex), like an ice cream cone or a traffic cone. Volume measures the space inside it, in cubic units.
Why does the volume of a cone matter?
It tells you how much a cone-shaped container can actually hold — and the answer is smaller than most people expect.
Fill a cone with radius 3 m and height 4 m, and you get about 37.7 m³. A cylinder with that exact same radius and height holds about 113.1 m³ — exactly three times as much. A cone is always exactly one-third of the matching cylinder.
The formula
| Symbol | What it means | Example |
|---|---|---|
| V | The volume you want to find | 1,130.97 m³ |
| π (pi) | A fixed number, about 3.14159 | 3.14159 |
| r | The radius of the circular base | 6 m |
| h | The height: straight-line distance from base to apex | 10 m |
How to find the volume step by step
- 1Measure the radius of the circular base.
- 2Measure the height. Straight up from the base to the pointed tip, not the slanted side.
- 3Square the radius, then multiply by π. This gives the base's area.
- 4Multiply that base area by the height.
- 5Divide the result by 3.
Now try it yourself. Change the radius and height in the calculator below and watch the volume update, with every step shown.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using the slant height instead of the vertical height. The h in this formula is the straight-up height, not the length of the sloped side.
- Confusing volume with surface area. Volume is what fits inside; for the paper or material needed to build the cone's outer surface, see the surface area of a cone calculator.
- Squaring the wrong measurement. Only the radius gets squared, never the height.
Tips and tricks
- Doubling the height doubles the volume; doubling the radius multiplies it by 4, since the radius is squared.
- To find the radius from the diameter, divide by 2 before squaring.
Where you'll use it in real life
- Food: estimating how much ice cream fits in a cone.
- Construction: working out how much sand or gravel is in a conical pile.
- Engineering: calculating the capacity of a cone-shaped funnel or hopper.
- Design: comparing storage capacity between conical and cylindrical containers.
Quick summary
To find the volume of a cone, square the radius, multiply by π and by the height, then divide by 3. A cone always holds exactly one-third as much as a cylinder with the same base and height. Use the calculator above to check your own cone instantly.
Worked example
A conical sand pile at a construction site has a radius of 6 meters and a height of 10 meters. How much sand does the pile contain?
- 01V = (1/3)πr²h
- 02V = (1/3)π(6)²(10)
- 03V = (1/3)π(360)
- 04V = 120π
- 05V = 376.99 m³
These numbers are pre-filled in the calculator above.
Frequently asked questions
Why is a cone's volume one-third of a cylinder's?
A cone tapers smoothly from its full base down to a single point, while a cylinder keeps the same circular cross-section all the way up. Geometrically, this works out to exactly one-third the volume for the same radius and height.
What's the difference between the height and the slant height of a cone?
The height is the straight vertical distance from the base to the tip. The slant height is the distance along the sloped outer surface, which is longer. Volume uses the vertical height, while [surface area](/calculator/surface-area-cone) uses the slant height.
How do I find the volume of a cone from the diameter?
Divide the diameter by 2 to get the radius, then use V = (1/3)πr²h as usual.
Does doubling the height double a cone's volume?
Yes, since height appears only once in the formula. Doubling the radius instead multiplies volume by 4, since the radius is squared.
