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Math & Statisticsgeometry

Cone Calculator

Cone volume and surface, including truncated cones. The curved surface uses the slant height rather than the vertical height, which is the mistake that makes a paper cone come out too small.

Also called: cone volume calculator, frustum calculator.

Volume
0.39

Volume 0.39, slant height 1.58, curved surface 2.48 and total surface 3.27. A full cone, exactly one third of the cylinder that encloses it.

Slant height
1.58
Curved surface area
2.48
Total surface area
3.27
Base area
0.79
Volume of the enclosing cylinder
1.18
On the shape
A full cone, exactly one third of the cylinder that encloses it.
How this is calculated
Method and background

How this is calculated

A cone is exactly one third of the cylinder that encloses it, which is shown alongside so the relationship is visible. The surface calculation is where errors happen: the curved face is a sector of a circle whose radius is the slant height, found from Pythagoras on the radius and vertical height. Using the vertical height instead makes the sheet too short. A truncated cone, or frustum, is the shape of most buckets and lampshades.

the curved surface uses the slant height, which is longer than the vertical height
s
Slant height
h
Vertical height

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a cone one metre across

Base diameter
1
Vertical height
1.5
Top diameter, for a truncated cone
0

Volume0.39

The volume is exactly a third of the enclosing cylinder

Open this example

a 3-4-5 triangle gives a clean slant

Base diameter
6
Vertical height
4
Top diameter, for a truncated cone
0

Volume37.7

boundary: Pythagoras on radius 3 and height 4

Open this example

Method and limits

What it assumes

  • A right circular cone with the apex above the centre of the base.

What it deliberately does not model

  • An oblique cone has the same volume but a different surface area, not computed here.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why is my paper cone too small?
Because the curved surface needs the slant height, not the vertical height. The slant is always longer, by Pythagoras on the radius and the height.
Is a cone really a third of a cylinder?
Exactly, for the same base and height. The same one-third relationship holds between a pyramid and its prism.