# cone

Dimensions of a cone.

Terms associated with a cone.

A cone is a solid geometrical figure (its name comes from the Greek *konos* for
pinecone) that has a circular or elliptical base and a vertex,
also known as an apex, lying outside the plane of the base and that is formed
from all the line segments joining points on the edge of the base to the
vertex.

If the base is a circle, the shape is a **circular
cone**; if the line, or **axis**, from the center of
the base to the vertex is perpendicular to the base, then it is a **right
cone** (an ice-cream cone is a right circular cone); otherwise it's
an **oblique cone**. The curved lateral surface of the cone
is called a **nappe**. If the cone is extended in both directions
from the vertex, the result is a **double cone** or **bicone**.
A section through a double cone that has been extended indefinitely in both
directions to form a **conic surface** is known as a conic
section.

Another way to think of a cone is as a surface
of revolution generated by a line that rotates around a fixed point,
at a fixed angle from another line (the axis), both lines passing through
that fixed point. The volume of a cone, of perpendicular height *h* and circular base of radius *r*, is 1/3π*r*^{ 2}*h*.

Take a solid cylinder of radius *r* and height 2*r*. Remove
the right double cone that passes through the center of the cylinder and
extends to meet the circular disks on the cylinder's top and bottom. Interestingly,
the volume of the remaining object and the volume of a sphere of radius *r* are the same.

### Puzzle

The Cone Puzzle (no. 202) from Henry Dudeney's *Amusements in Mathematics*^{1} runs as follows: "I have
a wooden cone. How am I to cut out of it the greatest possible cylinder?"

Solution: The simple rule is that the cone must be cut at one-third of its height.

## Reference

1. Dudeney, H. E. *Amusements in Mathematics*. New York: Dover,
1917. Reprinted Mineola, NY: Dover, 1958.