Cone Calculator

Base radius

Height

Result

Volume37.699112
Slant height5
Base area28.274334
Lateral surface47.12389
Total surface75.398224
Apex angle, °73.7398

One third of the cylinder that contains it

Volume, slant height from Pythagoras, lateral and total surface area, and the apex angle.

A cone holds exactly a third of the cylinder with the same base and height. Worth committing to memory — it saves learning a separate formula and explains where the three in the denominator comes from.

The slant height is not the height: it runs from the apex down the surface to the rim. Pythagoras gives it from the radius and height, and it is what the lateral area depends on.

What this calculator shows

Volume and slant height
Base area, lateral surface and total surface area
The apex angle, useful for funnels and for cutting patterns

What to keep in mind

The cone is right circular: the apex sits directly above the centre of the base.
Lateral area is πrl with l the slant height; unrolled it is a sector of a circle.

FAQs

Why is a cone one third of a cylinder?

It follows from integration or Cavalieri's principle. The practical demonstration is simpler: three cones of water fill one cylinder of the same base and height.

How does slant height differ from height?

Height is the perpendicular from apex to base; slant height runs along the surface to the rim. It is always longer and comes from Pythagoras.

How do I make a cone pattern?

Cut a sector of a circle with radius equal to the slant height and arc length equal to the base circumference. The sector angle is 360° × r / l.

What about a truncated cone?

That uses V = πh(R² + Rr + r²)/3 with two radii. This page covers the full cone only.

Worked example

A 3 by 4 cone

Input: Radius 3, height 4

Output: Volume 37.7, slant height exactly 5

Note: The slant comes out whole because 3, 4, 5 is a Pythagorean triple. The equivalent cylinder holds 113.1, precisely three times as much.

Useful links

All tools in Math & Statistics

Cone Calculator

Result

Related tools

Cylinder Calculator

Sphere Calculator

Pythagorean Theorem Calculator

Useful links