Inscribing and Circumscribing:

In the first diagram at the right, a prism is inscribed in a cylinder. A regular, right, pentagonal prism is inscribed in a right circular cylinder where the bases of the prism are inscribed in the bases of the cylinder and the lateral edges of the prism are touching the curved lateral surface of the cylinder.

Inscribed Prism

Circumscribed Prism

In the second diagram, shown above, the prism is circumscribed about the cylinder. The bases of the prism are circumscribed about the bases of the cylinder and the lateral faces of the prism are tangent to the lateral surface of the cylinder.

If the number of lateral faces (sides) of the prism is increased, the "space" between these two solids will decrease. As the number of lateral faces (sides) increases indefinitely (approaches infinity), the two solids will appear to blend into one solid (the cylinder).

If a prism whose bases are regular polygons is inscribed in, or circumscribed about, a right, circular cylinder, and if the number of lateral faces is indefinitely increased, the following "limiting concepts" will occur:

• the volume of the prism approaches the volume of the cylinder as a limit.

• the area of a cross section parallel to the bases of the prism approaches the area of a cross section parallel to the bases of the cylinder as a limit.

• the lateral area of the prism approaches the lateral area of the cylinder as a limit.

 So, can a regular prism with an extremely large number of lateral faces (sides) be called a cylinder ?

It certainly seems from theses observations that this "may" be the case. But the answer depends upon the definitions of prism, cylinder, polygon, and circle that are being followed.

One group of mathematical authors states that a prism is not a cylinder, adhering to the definition that the bases of a cylinder are curved surfaces and are not polygons with a large number of extremely tiny flat (straight) surfaces (sides).

Another group of authors defines a category of solids called "general cylinders" which include prisms as a form of cylinder.

Since the definition of the term "prism" references polygonal bases, this argument appears to be based upon the old two-dimensional argument as to whether a polygon with an infinite number of sides is a circle.

There does seem to be agreement that the term "cylinder" when used without qualification refers to a right circular cylinder.

At this level of study, we will be taking the position that a polygon is a finite set of straight line segments, and is not a circle. As such, a prism being defined as having polygonal bases, does not lend itself to being redefined as a cylinder. The fact that increasing the number of lateral faces of a regular prism creates a solid similar to a cylinder, will be viewed as a limiting concept and not as a redefinition of a cylinder.