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A cone is a three-dimensional geometric figure which consists of the set of all straight lines joining the boundary of a plane figure $PQR$ to a point $A$ not in the same plane of $PQR$:



The plane figure $PQR$ is called the base of the cone.


In the above diagram, the point $A$ is known as the apex of the cone.



Let a perpendicular $AE$ be dropped from the apex of a cone to the plane containing its base.

The length $h$ of the line $AC$ is the height of the cone.

Right Circular Cone

A right circular cone is a cone:

whose base is a circle
in which there is a line perpendicular to the base through its center which passes through the apex of the cone:
which is made by having a right-angled triangle turning along one of the sides that form the right angle.

In the words of Euclid:

When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone.
And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.

(The Elements: Book $\text{XI}$: Definition $18$)

Double Napped Cone

A double napped cone is a cone where the lines joining the apex to the circumference of the base extend indefinitely in either dimension: