About

Miscalculating cone volume leads to material waste in concrete pours, incorrect fill estimates for hoppers and silos, and flawed engineering specifications. The volume of a right circular cone is 13πr²h, where r is the base radius and h is the perpendicular height. This tool computes volume, lateral surface area, and total surface area using exact closed-form equations. It also derives the slant height l via the Pythagorean relationship when only r and h are known, or back-calculates h from l.

Inputs are validated against geometric constraints: slant height must exceed radius (l > r) and all dimensions must be positive. The calculator assumes a right circular cone with a flat circular base. Oblique cones or truncated cones (frustums) require different formulas not covered here. Results are rounded to six significant figures to avoid false precision.

Formulas

The volume of a right circular cone with base radius r and perpendicular height h:

V = 13 π r² h

The slant height l relates to r and h through the Pythagorean theorem:

l = √r² + h²

When the slant height l is known instead of h:

h = √l² − r²

Lateral (side) surface area:

A_L = π r l

Base area:

A_B = π r²

Total surface area:

A_T = π r ( r + l )

Where: V = volume, r = base radius, h = perpendicular height, l = slant height, A_L = lateral surface area, A_B = base area, A_T = total surface area, π ≈ 3.141592653589793.

Reference Data

Cone Type / Object	Typical Radius	Typical Height	Approx. Volume	Application
Ice cream cone (wafer)	2.5 cm	12 cm	78.5 cm³	Food industry
Traffic cone (standard)	14 cm	45 cm	9,236 cm³	Road safety
Sand pile (small)	0.5 m	0.4 m	0.105 m³	Bulk material storage
Volcanic cinder cone	250 m	300 m	1.96 × 10⁷ m³	Geology
Conical hopper (industrial)	1.2 m	1.8 m	2.714 m³	Grain / cement storage
Paper cup (conical)	3.5 cm	9 cm	115.5 cm³	Dispensers
Funnel (kitchen)	6 cm	10 cm	376.99 cm³	Liquid transfer
Roof turret (decorative)	1.5 m	2.5 m	5.89 m³	Architecture
Christmas tree (approx.)	0.8 m	2.0 m	1.34 m³	Decoration sizing
Conical flask (lab, 250 mL)	4.3 cm	13.5 cm	261 cm³	Chemistry labs
Pile of gravel (medium)	2 m	1.5 m	6.28 m³	Construction material
Speaker cone (subwoofer)	15 cm	8 cm	188.5 cm³	Audio engineering
Nose cone (model rocket)	2 cm	7 cm	29.3 cm³	Aerospace hobby
Tepee tent (approx.)	2.5 m	4 m	26.18 m³	Camping / shelter
Stockpile (coal, large)	10 m	6 m	628.3 m³	Energy / mining

Frequently Asked Questions

Volume scales with the square of the radius. Doubling r while holding h constant multiplies volume by 4, since V = 13πr²h. Replacing r with 2r gives 4πr²h/3.

The perpendicular height is derived as h = √l² − r². When l ≤ r, the expression under the radical becomes zero or negative, yielding a degenerate flat disk or an imaginary number. A valid three-dimensional cone requires l > r.

The volume formula 13πr²h is valid for oblique cones by Cavalieri's principle, provided h is the perpendicular distance from apex to base plane. However, the lateral surface area formula πrl applies only to right circular cones. Oblique cones require integration over the curved surface.

One liter equals 1000 cm³ or 0.001 m³. If the calculator outputs volume in cm³, divide by 1000. If in m³, multiply by 1000. For imperial, 1 in³ = 0.016387 L.

A cone's volume is exactly 13 of the volume of a cylinder with the same base radius and height. This factor of 3 is provable via integral calculus or Cavalieri's principle. Practically, this means you need three cones of water to fill a matching cylinder.

The semi-vertical angle α is the angle between the axis (height) and the slant surface. It is computed as α = arctan(r ÷ h). For a cone with r = 5 and h = 10, α ≈ 26.57°.