Mathematics, Statistics & Geometry

Volume of Cone Calculator

Calculate the exact 3D cubic capacity of any standard geometric cone using the radius of its circular base and its perpendicular vertical height.

Volume
314.159
Calculation StepsRadius (r) = 5, Height (h) = 12 Formula: V = (1/3)πr²h V = (1/3) * π * (5)² * 12 V = 1.0472 * 25 * 12 = 314.1593

Calculated locally in your browser. Fast, secure, and private.

The Geometry of Tapered Space

The Volume of Cone Calculator rapidly computes the 3D capacity of conic topologies. By geometrically linking the flat circular base area with the perpendicular apex height, it calculates exact cubic displacement.

V=13πr2h\begin{aligned} V = \frac{1}{3} \pi r^2 h \end{aligned}

Where:
V=
The total 3D cubic space contained within the cone
r=
The distance from the center of the flat circular base to its outer edge
h=
The straight perpendicular line from the center of the base to the top tip

The One-Third Rule

One of the most beautiful physical proofs in mathematics relates the Cone to the Cylinder. The formula for a cylinder is πr2h\pi r^2 h. The formula for a cone is 13πr2h\frac{1}{3} \pi r^2 h.

This is not a rough approximation; it is a mathematically perfect constant. If a manufacturer builds a cylindrical grain silo, and right next to it builds a conical hopper with the exact same height and radius, the cone will hold exactly 33.333...% of the cylinder's capacity. This exact 1/3 ratio applies to pyramids and rectangular prisms as well!

Real-World Applications

  • Civil Engineering: Calculating the exact volume of sand, gravel, or road salt stored in massive conical stockpiles before winter highway deployment.
  • Aerospace Design: Determining the exact cubic volume of fuel that can fit inside the conical nose cone of a supersonic jet or orbital rocket.
  • Manufacturing: Designing the exact geometric dimensions of a conical paper water cup so that it holds precisely 4 ounces of liquid without spilling.

Frequently Asked Questions

If you take a perfect cylinder and a perfect cone that have the exact same height and exact same base, it will take exactly 3 cones full of water to fill the cylinder perfectly. The cone is exactly 1/3 the volume of the cylinder.

Vertical height (h) goes straight down the middle like a tent pole. Slant height goes down the slanted outside canvas. You MUST use the straight vertical height to calculate Volume.

Mathematically, it does not matter. The volume of an upright traffic cone and an upside-down ice cream cone is calculated using the exact same formula.

Thanks to Cavalieri's Principle, an oblique (leaning) cone has the exact same volume as a perfectly straight right cone, as long as the vertical height and base radius are the same.

Volume is a 3D measurement, so the final answer must always be in cubic units (e.g., cubic centimeters, cubic inches, gallons, or liters).

Related Calculators in Mathematics, Statistics & Geometry

Angle Between Vectors Calculator

Try It Now

ANOVA Calculator

Try It Now

Antilog Calculator

Try It Now

Arc Length of Curve Calculator

Try It Now

Area of Annulus Calculator

Try It Now

Area of Circle Calculator

Try It Now