Spherical surface. Ball (sphere).
Center, radius and diameter of a ball.
Sections of a ball. Large circle. Archimedean theorem. Parts of
spherical segment, spherical layer, spherical zone, spherical
Spherical surface is a
geometrical locus in a space, that is a totality of all
points, equally removed from one point O, which is called a
center ( Fig.90 ).Radius AO and diameter AB are
defined as well as in a circumference.
Ball (sphere) is a body,
bounded by a spherical surface. It is possible to receive a
ball, rotating a half-circle (or a circle) about its diameter.
All plane sections of a ball – circles (Fig.90). The
largest circle is in a section, going through a center of a ball
and is called a large circle. Its radius is equal to a
radius of a ball. Any two large circles are intersected along a
diameter of a ball (AB, Fig.91). This diameter is
also a diameter for each of these intersecting circles. Through
two points of a spherical surface, placed on the ends of the
same diameter (A and B, Fig.91 ), it is possible to draw an
innumerable set of large circles. For instance, through the
poles of Earth it is possible to draw an infinite number of
A volume of a ball is less than a volume
of a curcumscribed cylinder ( Fig.92 ) in one and a half
times; and a surface of a ball is less than a full surface of
this cylinder also in one and a half times (Archimedean
S ball and
- a surface and a volume of a ball correspondingly;
and V cyl
- a full surface and a volume of a circumscribed cylinder.
Parts of a ball. A part of a
ball (sphere), cut out by any plane ( ABC, Fig.93 ), is
called a spherical segment. The circle ABC is called a base
of a spherical segment. The segment MN of a perpendicular, drawn
from a center N of the circle ABC till
intersection with a spherical surface, is called a height
of a spherical segment. The point M is called a vertex
of a spherical segment.
A part of a sphere, concluded between the two
parallel planes ABC and DEF, crossing the spherical surface (
Fig.93 ), is called a spherical layer; a curved
surface of a spherical layer is called a spherical zone.
Circles ABC and DEF are called bases of a
spherical zone. The distance NK between the bases of a spherical
zone is its height.
A part of a ball, bounded by a curved surface of a spherical
segment ( AMCB, Fig.93 ) and the conic surface OABC, a
base of which is a base of a segment ( ABC ), and a
vertex – a center of a ball O, is called a spherical