Volume of a Cone
What is the Volume of a Cone?
A cone is exactly one-third the volume of a cylinder that shares the same base and height — which explains the 1/3 factor built into the formula. Think of ice cream cones, party hats, or funnels as everyday examples of this shape.
A classic demonstration of this relationship: if you fill a cone with water and pour it into a cylinder with the same base and height, you need to repeat it exactly three times to fill the cylinder.
What Each Variable Means
When to Use It
- Finding the volume of any cone-shaped object
- Comparing a cone's capacity to a cylinder with the same dimensions
- As a building block in more complex composite-solid volume problems
Step-by-Step Example
Problem: An ice cream cone has base radius 4 cm and height 9 cm. What is its volume?
Radius and height are both given.
r = 4 cm, h = 9 cmSquare the radius first.
r² = 16Multiply by (1/3)π and the height.
V = (1/3) × π × 16 × 9 = 48πInteractive Calculator
Common Mistakes
Mistake: Forgetting the 1/3 factor and computing a cylinder's volume instead.
Fix: A cone's volume is exactly 1/3 of the cylinder with the same base and height — leaving out that factor triples the true volume.
Mistake: Using the slant height instead of the perpendicular height.
Fix: h in this formula is the perpendicular height from base to apex, not the slant length along the cone's curved surface — those are different measurements unless the cone is unusually shaped.
Practice Questions
A cone has radius 3 cm and height 12 cm. Find its volume.
A cone and a cylinder share the same base and height. If the cylinder's volume is 90 cm³, what is the cone's volume?
Hint: The cone is always exactly 1/3 the cylinder's volume.
Frequently Asked Questions
Why is a cone exactly 1/3 of a cylinder's volume?
This can be proven with calculus (integrating the area of circular cross-sections that shrink linearly to a point) or demonstrated experimentally by pouring water — both confirm the exact 1/3 ratio.
Does the formula change for an oblique (slanted) cone?
No — as long as r is the base radius and h is the perpendicular height (not the slant), the same formula V = (1/3)πr²h applies regardless of whether the cone leans.