Cone Volume Calculator

Enter the base radius and height of a right circular cone to find its volume, base area, slant height, and total surface area using V = (1/3)πr²h.

Quick Facts

Formula
V = (1/3) × π × r² × h
A cone holds exactly one third the volume of a cylinder with the same base radius and height.
Assumption
Right circular cone
Apex centered directly above the middle of a circular base; use radius, not diameter.

Your Results

Calculated
Cone volume
-
V = (1/3)πr²h
Base area
-
A = πr²
Slant height
-
l = √(r² + h²)
Total surface area
-
A = πr(r + l)

Ready

Set the base radius and height, then press Calculate.

Understanding Cone Volume

This tool computes the volume of a right circular cone — a cone whose circular base has its center directly beneath the apex (tip). Given the base radius and the height, it returns the volume along with three related measurements: the base area, the slant height, and the total surface area.

The formula

The volume of a right circular cone is:

V = (1/3) × π × r² × h

where r is the radius of the circular base and h is the perpendicular height from the base to the apex. This is exactly one third of the volume of a cylinder with the same radius and height (Vcylinder = πr²h) — a cone always occupies a third of the "can" that circumscribes it.

Two related quantities use the slant height, the distance along the cone's slanted side from the base edge to the apex, found with the Pythagorean theorem: l = √(r² + h²). The base area is πr², and the total surface area (base plus the curved lateral surface) is πr(r + l).

Common mistakes

  • Radius vs. diameter: the formula needs the radius (half the diameter). Entering the full diameter as if it were the radius quadruples the computed volume, since the formula squares r.
  • Height vs. slant height: h is the straight vertical distance to the apex, not the length along the cone's surface. Using the slant height in place of h overstates the volume for any cone that isn't perfectly flat.
  • Mixed units: keep radius and height in the same unit before entering them. The result is reported in that unit cubed (e.g., cm³), so mixing centimeters and meters produces a meaningless number.

Real-world applications

  • Estimating the capacity of ice cream cones, paper cups, funnels, and party hats
  • Sizing conical hoppers, silo bottoms, and material funnels in engineering
  • Estimating the volume of a conical pile of sand, gravel, or grain
  • Manufacturing and packaging design that specifies volume from a conical mold
  • Geometry coursework and checking 3D-modeling or CAD volume outputs

Frequently Asked Questions

What is the formula for the volume of a cone?
The volume of a right circular cone is V = (1/3) × π × r² × h, where r is the radius of the circular base and h is the perpendicular height from the base to the apex. A cone always holds exactly one third the volume of a cylinder that shares the same base radius and height.
Do I use radius or diameter in the formula?
The formula needs the radius, not the diameter. If you only measured across the base, divide that diameter by 2 before entering it as the radius. Using the diameter in place of the radius will overstate the volume by a factor of four, since the formula squares that value.
Is height the same as slant height?
No. Height (h) is the straight vertical distance from the base to the apex, measured perpendicular to the base. Slant height (l) is the distance along the cone's outer surface from the base edge to the apex, found with l = √(r² + h²). Volume uses height, not slant height; surface area uses slant height.
Does this formula work for a truncated cone or frustum?
No. This calculator assumes a complete right circular cone coming to a single point (apex). A frustum, a cone with the tip cut off, has two different radii and needs a separate frustum volume formula: V = (1/3) × π × h × (r1² + r1 × r2 + r2²).

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