Volume of a Cone Calculator
Calculate the volume, surface area, slant height, and lateral area of a cone from its radius and height. See also Volume of Cylinder Calculator and Volume of Pyramid Calculator.
How to Calculate the Volume of a Cone
To find the volume of a cone, measure the radius of the circular base and the perpendicular height from the base to the apex. Square the radius, multiply by π and the height, then divide by 3. A cone is essentially one-third of a cylinder with the same base and height. This calculator also computes the slant height, total surface area, and lateral surface area.
Cone Volume Formula
V = (1/3) × π × r² × h
Slant Height: l = √(r² + h²)
Lateral Area = π × r × l
Total SA = π × r × (r + l)
Base Area = π × r²
Example
Find the volume of a cone with radius 4 and height 8:
V = (1/3) × π × r² × h
V = (1/3) × π × 4² × 8
V = (1/3) × π × 128
V ≈ 134.0413 cubic units
Slant height l = √(16 + 64) = √80 ≈ 8.9443
Cone Volume Reference Table
| Radius | Height | Volume | Surface Area |
|---|---|---|---|
| 1 | 3 | 3.1416 | 13.0762 |
| 2 | 4 | 16.7552 | 40.6656 |
| 3 | 5 | 47.1239 | 83.2298 |
| 4 | 6 | 100.5310 | 140.8829 |
| 4 | 8 | 134.0413 | 162.6625 |
| 5 | 8 | 209.4395 | 226.7284 |
| 5 | 10 | 261.7994 | 254.1602 |
| 6 | 8 | 301.5929 | 301.5929 |
| 6 | 12 | 452.3893 | 365.9907 |
| 8 | 10 | 670.2064 | 522.9181 |
| 8 | 15 | 1005.3096 | 628.3185 |
| 10 | 10 | 1047.1976 | 758.4476 |
| 10 | 15 | 1570.7963 | 880.5179 |
| 10 | 20 | 2094.3951 | 1016.6407 |
| 15 | 20 | 4712.3890 | 1884.9556 |
Frequently Asked Questions
What is the volume of a cone?
The volume of a cone is the total three-dimensional space enclosed within its circular base and curved surface tapering to a point (apex). It equals one-third of the volume of a cylinder with the same base and height.
What is the slant height of a cone?
The slant height is the distance from any point on the edge of the circular base to the apex, measured along the surface. It is calculated as l = √(r² + h²) using the Pythagorean theorem.
Why is the cone volume one-third of a cylinder?
This can be proven using calculus (integration) or Cavalieri's principle. Intuitively, three cones with the same base and height can fill exactly one cylinder of the same dimensions.
What is the difference between lateral area and total surface area?
The lateral area is only the curved surface of the cone (πrl). The total surface area adds the circular base (πr²), giving πr(r + l).
Does this formula work for oblique cones?
The volume formula V = (1/3)πr²h works for oblique cones as long as h is the perpendicular height. The surface area and slant height formulas only apply to right circular cones.