Volume of a Cone
Calculate the volume of a cone using V = (1/3)πr²h.
A cone holds exactly one-third the volume of a cylinder with the same base and height.
The Formula
V = (1/3) × π × r² × h
The volume of a cone is exactly one-third the volume of a cylinder with the same radius and height.
Variables
| Symbol | Meaning |
|---|---|
| V | Volume of the cone |
| π | Pi, approximately 3.14159 |
| r | Radius of the circular base |
| h | Height of the cone (measured perpendicular to the base) |
Example 1
Find the volume of a cone with radius 5 cm and height 12 cm
V = (1/3) × π × r² × h = (1/3) × π × 5² × 12
V = (1/3) × π × 25 × 12 = (1/3) × π × 300
V = 100π
V ≈ 314.16 cm³
Example 2
An ice cream cone has a diameter of 6 cm and a depth of 11 cm. What is its volume?
Diameter = 6 cm, so radius = 3 cm, height = 11 cm
V = (1/3) × π × 3² × 11 = (1/3) × π × 9 × 11
V = (1/3) × 99π = 33π
V ≈ 103.67 cm³
When to Use It
Use the volume of a cone formula when:
- Calculating the capacity of cone-shaped containers (funnels, ice cream cones)
- Working with conical piles of material (sand, gravel)
- Solving geometry problems involving cones
- Comparing cone volumes to cylinder volumes (a cone is always 1/3 of the equivalent cylinder)
Key Notes
- Formula: V = ⅓πr²h: The base area is πr² (same as a cylinder); the ⅓ factor reflects that a cone is exactly one-third the volume of the cylinder with the same base and height. This can be proven by integration or by the 3-cone filling demonstration.
- Surface area: Lateral surface area = πrl where l = √(r² + h²) is the slant height. Total surface area (including base) = πrl + πr². These are separate quantities — calculate which one the problem needs.
- Oblique vs right cones: An oblique cone has an apex directly above a point other than the center. Cavalieri's principle proves its volume is still ⅓πr²h (using the perpendicular height), identical to a right cone with the same base and height.
- Frustum (truncated cone): When the apex is cut off, V = ⅓πh(R² + Rr + r²) where R and r are the radii of the two circular bases. This formula appears in engineering for tapered pipes and hopper bins.
- Applications: Volume of cone formulas are used to calculate material for conical structures (funnels, silos, volcano models), ice cream cone capacity, and the shape of milling machine cutters.