About the Cone Calculator
This cone calculator solves a right circular cone from two measurements. Give the base radius and either the vertical height or the slant height, and it returns the volume, the missing height, the lateral (side) surface area, the base area, the total surface area and — when you choose a real unit — the capacity in liters.
It is handy for geometry homework and for practical shapes: the volume of an ice-cream cone, a funnel, a conical hopper or a pile of sand or gravel, the amount of sheet material needed to roll a cone, or the capacity of a conical tank bottom.
The height must be the perpendicular height from the base to the tip, not the sloping side; if you only know the sloping side, choose the slant-height option and the calculator works out the vertical height for you.
With the default inputs, the volume is 37.6991. Change any value above to recalculate instantly.
How to use the cone calculator
- 1Choose whether you know the vertical height or the slant height.
- 2Select your unit.
- 3Enter the base radius (half the diameter).
- 4Enter the height or slant height.
- 5Read the volume, surface areas and capacity.
Formula and method
A cone holds exactly one third of the cylinder with the same base and height, so its volume is ⅓ × base area × height = ⅓πr²h. The radius, perpendicular height and slant height form a right triangle, so the slant height follows from the Pythagorean theorem, l = √(r² + h²), and the height from h = √(l² − r²).
Unrolled, the curved side of a cone is a sector of a circle with radius l and arc length 2πr, giving a lateral area of πrl. Adding the circular base πr² gives the total surface area πr(r + l). Capacity uses 1 m³ = 1,000 liters.
- r
- Radius of the circular base
- h
- Perpendicular height from base to apex
- l
- Slant height along the side
- V
- Volume
Worked examples
Cone with radius 3 cm and height 4 cm
The slant height is √(9 + 16) = 5. Volume = ⅓ × π × 9 × 4 = 12π ≈ 37.70 cm³; the side area is π × 3 × 5 ≈ 47.12 cm² and the base adds 28.27 cm².
Conical pile: radius 5 ft, slant height 13 ft
The height is √(169 − 25) = 12 ft, so V = ⅓ × π × 25 × 12 = 100π ≈ 314.16 ft³ — about 8,896 liters.
Ice-cream cone: radius 2.5 cm, height 11 cm
⅓ × π × 2.5² × 11 ≈ 72 cm³, so the cone holds about 72 mL when filled level to the rim.
Frequently asked questions
What is the formula for the volume of a cone?+
V = ⅓πr²h, where r is the base radius and h is the perpendicular height. It is one third of the volume of a cylinder with the same base and height.
How do you find the slant height of a cone?+
Use the Pythagorean theorem on the radius and height: l = √(r² + h²). A cone with r = 3 and h = 4 has a slant height of 5.
What is the surface area of a cone?+
The lateral (curved) area is πrl and the base is πr², so the total surface area is πr(r + l). For an open cone, like a funnel, use only the lateral area.
How do I find the volume of a cone from its diameter?+
Halve the diameter to get the radius, then use V = ⅓πr²h. A cone 10 cm across and 12 cm high has r = 5 and a volume of about 314 cm³.