Circles have long intrigued humans. Often called the perfect
the circle occurs frequently in engineering design, art,
architecture, and applied
mathematics. Almost 2200 years ago, Archimedes sought to find a
the same area as a given circle. This problem is called
squaring the circle.
The circle forms the base for systems of navigation on the sea
and in the air. The
geometry of circles involves not only the relationships of the
principal parts of
the circle but also the relation of the circle to lines, angles,
Circles are symbolic of infinity, eternity,
and completeness. They appear in the literature and architecture of many
is the set of
all points in a plane equidistant from a given point. This distance is the
length of the
and the given
point is the circle’s
. A circle is named by its center point. For example, a circle
with center at point
referred to as “circle
Figure 19.1 is a drawing of a circle with nearby points. Note
that a point may lie either
circle; or it may lie
the circle, making it an
of the circle; or it may lie
the circle, making it an
of the circle. A common mistake
is to view interior points, including the center, of a circle as part of the
circle. The circle consists of
points a given distance from the center.
The plural of “radius” is “radii.”
is any line
segment joining the center of a circle with a point on the circle’s
is a line
segment joining any two points on the circle’s circumference.
is any line
segment through the center of the circle joining two points on the
circumference. The diameter’s length is twice the length of the radius. Thus,
the diameter is the longest chord that can be drawn in a circle.
The words “radius,” “diameter,” and “circumference”
can mean either these physical aspects of the circle or the specific
measurement assigned to each.
An angle formed by two radii is a
. We say that a chord or an angle
an arc, or an arc is
by an angle or a chord.
is a portion of
the circle lying between two points on the circle. A chord’s endpoints split
the circle into two arcs, the
, as shown in Figure 19.2. Since the diameter is the longest
chord in a circle, it subtends two arcs equal in measure. These arcs are
Arcs of a
is an arc
measuring one-half the circumference of a circle, a half circle.
A wedge-of-pie shaped area bounded by an arc and its two related
radii is a
The area bounded by an arc and its chord is a
is a line that
intersects the circle at two points. The part of a secant that is not outside
the circle is a chord.
A line that intersects a circle at one and only one point is
. Note that a tangent does not
cut through the circle. Referring to Figure 19.3, a tangent at point
is perpendicular to the radius
. In fact, it is the only line
perpendicular to radius
line and radi
circle has its vertex on the circle and its sides in the interior of the
circle. It subtends an arc, as shown in Figure 19.4 where angle
is an inscribed angle and angle
is the corresponding
. In fact, angle
is said to be the central angle
by inscribed angle
. Figure 19.4 illustrates an
angle inscribed in a circle.
An inscribed angle
The important geometric features of a circle are summarized in
Important geometric features
of a circle
are circles with a common center
but different radii, as shown in Figure 19.6.
circles are shown in Figures 19.7
Externally tangent circles
Internally tangent circles
A polygon is said to be
in a circle if all its sides are chords of a circle. This is the
same as requiring that each vertex of the polygon be a point on the circle, as
shown in Figure 19.8A. We could also say that the circle
hexagon inscribed in a circle
hexagon circumscribed on a circle
A polygon is
on a circle if all of its sides are tangent to the circle. A
circumscribed polygon is shown in Figure 19.8B. The circle is
in the polygon. Although regular
hexagons are shown in Figures 19.8A and 19.8B, an inscribed or circumscribed
polygon does not have to be regular. Figure 19.9 shows a nonregular
quadrilateral inscribed in a circle.
nonregular quadrilateral inscribed in a circle
An ellipse is shown in Figure 19.10. Like a circle, an ellipse
is a closed curve. Unlike a circle, which has a single
, its center, an ellipse has two
focal points, called
, making it appear flattened, or
egg shaped. An ellipse can be drawn by positioning pins at two points,
2, and loosely tying a length of
string between these points, the foci. Using a pencil point to hold the string
taut, follow the string to form the shape shown in Figure 19.10. Notice that
the distance remains constant.
The ellipse also has a center. It is the point of intersection
of the major and minor axes. These are special line segments that intersect at
right angles in the el
The major axis runs through the foci and center; the minor axis is the
perpendicular bisector of the major axis.
A circle is a special kind of ellipse
in which the two focal points are coincident.
features of an ellipse
Some examples of ellipses in technical fields are:
• The shape of the end section on a pipe that has been cut on a
• The projection of a circle in an isometric view of a part to
• The orbits of the planets around the sun.
Circles and ellipses are examples of
. The other conic sections are the
parabola and the hyperbola. Conic sections are the shapes obtained by passing a
plane through a right circular cone or cones at various angles.
Define the terms:
What is an inscribed polygon?
What is a circumscribed polygon?
many foci does an ellipse have?
an ellipse whose major and minor axes are equal. What figure has been drawn?
Copyright © 2006
Industrial Press Inc.