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. The one-third relationship with the cylinder is the whole trick to remembering the formula.
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.
- •The slant height runs along the sloped side, sqrt(r squared + h squared).
- •Lateral surface area is pi r times the slant height, and the base adds pi r squared more.
How to Use
- 1Enter the base radius of the cone.
- 2Enter the vertical height, straight up from base center to tip.
- 3Read the volume, with slant height and surface areas below.
- 4If you measured the sloped side instead of the vertical height, convert first: h = sqrt(slant squared - r squared).
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. For radius 4 and height 9, the base area is 50.27, the cylinder would hold 452.39, and the cone holds 150.80. The slant height comes from the Pythagorean theorem on the radius and height, here sqrt(16 + 81) = 9.85.
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.
Ice cream cone, radius 3 cm, height 12 cm
About 113 ml of ice cream fits inside, before the scoop on top.
Gravel pile 2 m tall with a 3 m radius
Loose material settles into a cone, and this is how yards of gravel get estimated.
Cone Volumes for Common Dimensions
Volume and slant height by radius and height.
| Radius | Height | Volume | Slant height |
|---|---|---|---|
| 2 | 6 | 25.13 | 6.32 |
| 3 | 4 | 37.70 | 5.00 |
| 4 | 9 | 150.80 | 9.85 |
| 5 | 12 | 314.16 | 13.00 |
| 6 | 8 | 301.59 | 10.00 |
Common mistakes
- Dropping the one third. Base area times height gives the surrounding cylinder, 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, so this mistake quadruples the volume.