Basic Mathematics for College Students

9.8 Circles 793

A chordof a circle is a line segment that connects two points on the circle. A
diameteris a chord that passes through the center of the circle. Since a diameter of
a circle is twice as long as a radius , we have

Each of the previous definitions is illustrated in figure (a) below, in which is the
center of the circle.

O

D 2 r

r

D

A

O

E

B

C

D

Diameter

CD

Chord

AB

Radius

OE

O

C

B

A

E

D

(a) (b)

Any part of a circle is called an arc.In figure (b) above, the part of the circle from
point to point that is highlighted in blue is AB, read as “arc .”CDis the part
of the circle from point to point that is highlighted in green. An arc that is half of
a circle is a semicircle.

C D

A B AB

Semicircle

A semicircleis an arc of a circle whose endpoints are the endpoints of a

diameter.

If point is the center of the circle in figure (b), is a diameter and AED is a
semicircle. The middle letter distinguishes semicircle AED (the part of the circle
from point to point that includes point ) from semicircle ABD (the part of the
circle from point to point that includes point ).
An arc that is shorter than a semicircle is a minor arc.An arc that is longer than
a semicircle is a major arc.In figure (b),

AEis a minor arc and ABEis a major arc.

A D B

A D E

E

O AD

Success Tip It is often possible to name a major arc in more than one way.

For example, in figure (b), major arc ABEis the part of the circle from point

to point that includes point. Two other names for the same major arc are

ACEand ADE.

E B

A

2 Find the circumference of a circle.

Since early history, mathematicians have known that the ratio of the distance around
a circle (the circumference) divided by the length of its diameter is approximately 3.
First Kings, Chapter 7, of the Bible describes a round bronze tank that was 15 feet from
brim to brim and 45 feet in circumference, and. Today, we use a more precise
value for this ratio, known as (pi). If is the circumference of a circle and is the
length of its diameter, then

p where p3.141592653589...^227 and 3.14 are often used as estimates of .p

C

D

p C D

4515  3