Calculate cone volume from the radius (or diameter) and height of a cone. This calculator also rearranges the volume formula to solve for the radius or height when you already know the volume, reports the slant height and surface area, and handles truncated cones (frustums).
Cone Volume Formula
The volume of a cone is one third of the volume of a cylinder with the same base and height:
V = (1/3) * pi * r^2 * h
If you know the diameter instead of the radius, substitute r = d / 2. Rearranging the same formula lets you solve for the other inputs:
r = sqrt( 3V / (pi * h) ) h = 3V / (pi * r^2)
The slant height and surface area use the right triangle formed by the radius, the height, and the side of the cone:
l = sqrt( r^2 + h^2 ) A_lateral = pi * r * l A_total = pi * r * (r + l)
For a truncated cone (a frustum) with a bottom radius R, a top radius r, and a height h between the two circular faces:
V = (1/3) * pi * h * (R^2 + R*r + r^2)
- V = volume of the cone
- r = base radius (top radius for a frustum)
- R = bottom radius of a frustum
- d = base diameter, equal to 2r
- h = vertical height, measured straight up from the base to the apex
- l = slant height, the distance up the sloped side
- pi = 3.14159..., the ratio of a circle's circumference to its diameter
In the default mode the calculator takes a radius or diameter and a height and returns the volume. The radius and height modes invert that calculation so you can supply a known volume and recover the missing dimension. The cone details mode adds the slant height and surface area. The truncated cone mode uses the frustum formula, which reduces to the standard cone volume when the top radius is zero.
Typical Cone Dimensions and Volumes
The table below shows the volume produced by a few radius and height pairs so you can sanity check a result. All volumes use V = (1/3) * pi * r^2 * h.
| Radius | Height | Volume |
|---|---|---|
| 2 | 6 | 25.13 |
| 3 | 5 | 47.12 |
| 5 | 10 | 261.80 |
| 7 | 12 | 615.75 |
| 10 | 10 | 1047.20 |
The next table converts a finished volume into common units. One cubic foot equals 1728 cubic inches, and one US gallon equals 231 cubic inches.
| From | To | Multiply by |
|---|---|---|
| cubic inches | cubic feet | 0.000579 |
| cubic inches | US gallons | 0.004329 |
| cubic feet | US gallons | 7.4805 |
| cubic cm | liters | 0.001 |
Example Problems
Example 1. A cone has a base radius of 4 and a height of 9. Square the radius to get 16, multiply by the height to get 144, multiply by pi to get about 452.39, then divide by 3. The volume is about 150.80.
Example 2. A truncated cone has a bottom radius of 6, a top radius of 3, and a height of 8. The bracket is R^2 + R*r + r^2 = 36 + 18 + 9 = 63. Multiply by the height to get 504, multiply by pi to get about 1583.36, then divide by 3. The volume is about 527.79.
Frequently Asked Questions
Do I use the slant height or the vertical height in the volume formula? Use the vertical height, measured straight up from the center of the base to the tip. The slant height runs along the sloped surface and is always longer. If you only have the slant height and the radius, find the vertical height first with h = sqrt(l^2 - r^2), then use it in the volume formula.
What if I measured the diameter instead of the radius? Divide the diameter by two to get the radius, then use the formula as written. The calculator lets you enter a diameter directly and does this step for you.
Why is a cone exactly one third of a cylinder? A cone and a cylinder that share the same base radius and the same height enclose volumes that differ by a fixed factor of three. Three cones of identical base and height fill the matching cylinder exactly, which is why the formula carries the 1/3 term.
