Volume of a cone

Definition: The number of cubic units that will exactly fill a cone.
Try this Drag the orange dots to adjust the radius and height of the cone and note how the volume changes.

The volume enclosed by a cone is given by the formula Where r is the radius of the circular base of the cone and h is its height. In the figure above, drag the orange dots to change the radius and height of the cone and note how the formula is used to calculate the volume.

Oblique cones

Recall that an oblique cone is one that 'leans over' - where the apex is not over the base center point. Drag the apex left and right above to see this. It turns out that the volume formula works just the same for these. You must remember to use the perpendicular height in the formula.

To illustrate this, in the figure above, click 'Reset' then 'Freeze height'. As you drag the apex left and right, watch the volume calculation and note that the volume never changes.

Relation to a cylinder

A cone embedded in its circumscribed cylinder Recall that the volume of a cylinder is

If you compare the two formulae, you will see one is exactly a third of the other. This means that the volume of a cone is exactly one third the volume of the cylinder with the same radius and height.

Such a cylinder is the "circumscribed cylinder" of the cone - the smallest cylinder that can contain the cone. In the figure above, select "Show cylinder" to see the cone embedded in its circumscribed cylinder.

Relation to pyramid

The volume of a cone and a pyramid are calculated in a similar way. They are both equal to one third the base area times the height. See Volume of a pyramid. In fact, you can think of a cone as a pyramid with an infinite number of sides. To see this go to Pyramid definition and keep increasing the number osides. It will begin to look a lot like a cone.

Things to try

  • In the figure above, click "hide details".
  • Drag the orange dots to set the radius and height of the cone.
  • Calculate the volume of the cone using the formula
  • Click "show details" to check your answer.
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