Chapter 4: Geometry

For a frustum of a right circular cone (Figure 4.37, bottom right),
Ô
Ð ´Ö ½ Ö ¾ µ ¾ · ¾
Lateral area ´Ö ½ · Ö ¾ µÐ
Total area ´Ö ¾ ½ · Ö¾ ¾ ·´Ö ½ · Ö ¾ µÐµ
Volume ½ ¿ ´Ö¾ ½ · Ö¾ ¾ · Ö ½Ö ¾ µ
and
4.17 SURFACES OF REVOLUTION: THE TORUS
A surface of revolution is formed by the rotation of a planar curve about an axis in
the plane of the curve and not cutting the curve. The Pappus–Guldinus theorem says
that:
1. The area of the surface of revolution on a curve is equal to the product of
the length of and the length of the path traced by the centroid of (which
is ¾ times the distance from this centroid to the axis of revolution).
2. The volume bounded by the surface of revolution on a simple closed curve
is equal to the product of the area bounded by and the length of the path
traced by the centroid of the area bounded by .
When is a circle, the surface obtained is a circular torus or torus of revolution
(Figure 4.38). Let Ö be the radius of the revolving circle and let Ê be the distance
from its center to the axis of rotation. The area of the torus is ¾ ÊÖ, and its volume
is ¾ ¾ ÊÖ ¾ .
FIGURE 4.38
A torus of revolution.
Ê
Ö
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