Similarly, an octahedron may be inscribed in a cube, and by letting it expand a little, the faces of the cube will cut off the corners of the octahedron. This is seen in the following figures:
This is a form that is found in crystals, and the computation of the surface and volume is an interesting exercise. The quartz crystal, an hexagonal pyramid on an hexagonal prism, is found in many parts of the country, or is to be seen in the school museum, and this also forms an interesting object of study in this connection.
The properties of the cylinder are next studied. The word is from the Greek kylindros, from kyliein (to roll). In ancient mathematics circular cylinders were the only ones studied, but since some of the properties are as easily proved for the case of a noncircular directrix, it is not now customary to limit them in this way. It is convenient to begin by a study of the cylindric surface, and a piece of paper may be curved or rolled up to illustrate this concept. If the paper is brought around so that the edges meet, whatever curve may form a cross section the surface is said to inclose a cylindric space. This concept is sometimes convenient, but it need be introduced only as necessity for using it arises. The other definitions concerning the cylinder are so simple as to require no comment.
The mensuration of the volume of a cylinder depends upon the assumption that the cylinder is the limit of a certain inscribed or circumscribed prism as the number of sides of the base is indefinitely increased. It is possible to give a fairly satisfactory and simple proof of this fact, but for pupils of the age of beginners in geometry in America it is better to make the assumption outright. This is one of several cases in geometry where a proof is less convincing than the assumed statement.
Theorem. The lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder by an element.
For practical purposes the cylinder of revolution (right circular cylinder) is the one most frequently used, and the important formula is therefore l = 2πrh where l = the lateral area, r = the radius, and h = the altitude. Applications of this formula are easily found.
Theorem. The volume of a circular cylinder is equal to the product of its base by its altitude.
Here again the important case is that of the cylinder of revolution, where v = πr2h.
The number of applications of this proposition is, of course, very great. In architecture and in mechanics the cylinder is constantly seen, and the mensuration of the surface and the volume is important. A single illustration of this type of problem will suffice.