12
Geometry
Copyright © Cengage Learning. All rights reserved.
12.5
Circles
Copyright © Cengage Learning. All rights reserved.
Circles
A circle is a plane curve consisting of all points at a given
distance (called the radius, r) from a fixed point in the
plane, called the center. (See Figure 12.44.)
The diameter, d, of the circle
is a line segment through the
center of the circle with endpoints
on the circle.
Note that the length of the
diameter equals the length
of two radii—that is, d = 2r.
Circle
Figure 12.44
3
Circles
The circumference of a circle is the distance around the
circle.
The ratio of the circumference of a circle to the length of its
diameter is a constant called  (pi).
The number  cannot be written exactly as a decimal.
Decimal approximations for  are 3.14 or 3.1416. When
solving problems with , use the  key on your calculator.
4
Circles
The following formulas are used to find the circumference
and the area of a circle.
C is the circumference and A is the area of a circle; d is the
length of the diameter, and r is the length of the radius.
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Example 1
Find the area and the circumference of the circle shown in
Figure 12.45.
Figure 12.45
The formula for the area of a circle given the radius is
A = r 2
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Example 1
cont’d
A =  (16.0 cm)2
A = 804 cm2
The formula for the circumference of a circle given the
radius is
C = 2 r
C = 2 (16.0 cm)
= 101 cm
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Circles
An angle whose vertex is at the center of a circle is called a
central angle.
Angle A in Figure 12.46 is a central angle.
Central angle
In general,
Figure 12.46
8
Common Terms and Relationships of a Circle
9
Common Terms and Relationships of a Circle
A chord is a line segment that has its endpoints on the
circle.
A secant is any line that intersects a circle at two points.
A tangent is a line that has only one point in common with
a circle and lies totally outside the circle.
In Figure 12.47, C is the center.
is a chord. Line n is a secant.
Line m is a tangent.
is a
diameter.
is a radius.
Figure 12.47
10
Arcs
11
Arcs
An inscribed angle is an angle whose vertex is on the
circle and whose sides are chords. The part of the circle
between the two sides of an inscribed or central angle is
called the intercepted arc.
In Figure 12.48, C is the center and
ACB is a central angle. DEF is an
inscribed angle.
is the intercepted
arc of ACB.
is the intercepted arc
of DEF.
Arcs of a circle
Figure 12.48
12
Arcs
The following three relationships are often helpful to solve
problems:
• The measure of a central angle in a circle is equal to the
measure of its intercepted arc. (See Figure 12.49.)
• The measure of an inscribed angle in a circle is equal to
one-half the measure of its intercepted arc.
(See Figure 12.49.)
Figure 12.49
13
Arcs
• The measure of an angle formed by two intersecting
chords in a circle is equal to one-half the sum of the
measures of the intercepted arcs. (See Figure 12.50.)
Figure 12.50
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Other Chords and Tangents
15
Other Chords and Tangents
A diameter that is perpendicular to a chord bisects the
chord. (See Figure 12.51.)
Figure 12.51
16
Other Chords and Tangents
A line segment from the center of a circle to the point of
tangency is perpendicular to the tangent.
(See Figure 12.52.)
Figure 12.52
17
Other Chords and Tangents
Two tangents drawn from a point outside a circle to the
circle are equal.
The line segment drawn from the center of the circle to this
point outside the circle bisects the angle formed by the
tangents. (See Figure 12.53.)
Figure 12.53
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