How to find the volume of a cone
The volume of a cone is one-third pi times radius squared times height: V = (1/3) × π × r² × h. With radius 3 and height 5, V = (1/3) × π × 9 × 5 = (1/3) × π × 45 ≈ 47.12 cubic units. A cone holds exactly one-third of a cylinder with the same base and height.
That one-third is not arbitrary. Stacking thin discs from the tip down to the base, each disc's area shrinks with the square of its distance from the apex, and adding them all up gives exactly a third of the enclosing cylinder. A cone holds exactly one third of the cylinder that shares its base and height, as you can check with the cylinder volume calculator.
How to use this calculator
Enter the base radius and the height (straight up from the base to the tip) in the same unit. The volume comes out in cubic units. For a round ball instead, use the sphere volume calculator.
Keep both measurements in one unit before you start — mixing centimetres with metres is the usual source of wildly wrong answers. Note also that the radius is squared, so it dominates the result: widening a cone has far more effect on its capacity than making it taller by the same amount.
Worked example
An ice-cream cone has a 3 cm radius and is 5 cm tall. Volume = (1/3) × π × 3² × 5 ≈ 47.12 cubic centimetres. Compare the cube shape with the cube volume calculator.
Since 1 cubic centimetre is 1 millilitre, that cone holds about 47 ml.
Frequently asked questions
- What is the formula for the volume of a cone?
- V = (1/3) × π × r² × h, where r is the base radius and h is the height.
- What is the volume of a cone with radius 3 and height 5?
- About 47.12 cubic units ((1/3) × π × 45).
- Why is there a 1/3 in the formula?
- A cone fills exactly one-third of a cylinder that shares the same base radius and height.
- What units is the volume in?
- Cubic units of whatever you measured the radius and height in.
- Should I use the slant height or the vertical height?
- Use the vertical height — the straight distance from the base to the tip, measured at a right angle to the base. The slant height runs along the sloping side and is always longer, so using it inflates the volume.