Chapter 4: Geometry

FIGURE 4.37
Top: a cone with vertex Ç and directrix . Bottom left: a right circular cone. Bottom right:
A frustum of the latter.
Ç
È
Ð
Ö
Ö ½
Ö ¾
Ð
If is a simple closed curve, we also apply the word cone to the solid enclosed
by the surface generated in this way (Figure 4.37, top). The volume contained between
È and the vertex Ç is
Î ½ (4.16.1)
¿
where is the area in the plane È enclosed by and is the distance from Ç and
È (measured perpendicularly).
The solid contained between È and a plane È ¼ parallel to È (on the same side
of the vertex) is called a frustum. It’s volume is
Î ½ ´ Ô
¿ · ¼ · ¼ µ (4.16.2)
where and ¼ are the areas enclosed by the sections of the cone by È and È ¼ (often
called the bases of the frustum), and is the distance between È and È ¼ .
The most important particular case of a cone is the right circular cone (often
simply called a cone). If Ö is the radius of the base, is the altitude, and Ð is the
length between the vertex and a point on the base circle (Figure 4.37, bottom left),
the following relationships apply:
Ô
Ð Ö¾ · ¾
Lateral area ÖÐ Ö Ô Ö ¾ · ¾
Ô
Total area Ö´Ð · Öµ Ö´Ö · Ö¾ · ¾ µ and
Volume ½ ¿ Ö¾
The implicit equation of this surface can be written Ü ¾ · Ý ¾ Þ ¾ ; see also Section
4.18.
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