Chapter 10: Area

Chapter 10:AreaSection 10-6:

Circles and Arcs

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Objectives:

To find the measures of central angles and arcs.

To find the circumference and arc length.







Vocabulary

Circle

Center

Radius

Congruent Circles

Diameter

Central Angle

Semicircle

Minor Arc

Major Arc

Adjacent Arcs

Circumference

Pi

Concentric Circles

Arc Length

Congruent Arcs







Circle

In a plane, a circle is the set of all points equidistance from a center point.







Center

The center is the point used to name the circle.







Radius

A radius is a segment that has one endpoint at the center and the other endpoint on the circle.







Congruent Circles

Congruent circles have congruent radii.







Diameter

A diameter is a segment that contains the center of a circle and has both endpoints on the circle.







Central Angle

A central angle is an angle whose vertex is the center of the circle.







ExampleTo learn how people really spend their time, a research firm studied the hour by hour activities of 3600 people. The participants were between 18 and 90 years old. Each participant was sent a 24-hour recording sheet every March for three years, from 2000-2002. The results are displayed in the chart.

What measure, in degrees, of the central angle is used for entertainment?







Arc

An arc is part of a circle.

We have three types of arcs:Semicircles-half a circle, exactly 180º

Minor Arcs- a minor arc is less than 180º

Major Arcs- a major arc is greater than 180º







Identifying Arcs

Name all the minor arcs.

Name all semicircles.

Name all the major arcs.







Adjacent Arcs

Adjacent arcs are arcs of the same circle that have exactly one point in common.

You can add the measures of adjacent arcs just like you can with adjacent angles.







Postulate 10-1:“Arc Addition Postulate”

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.







Finding Measures of Arcs

Find the measure of the following arcs:

1)

2)

3)

4)

mBC

mBD

mABC

mAB







Circumference

The circumference is the distance around the circle.







Pi

pThe number pi is the ratio of the circumference of a circle to its diameter.







Theorem 10-9:“Circumference of a Circle”

The circumference of a circle is p times the diameter.







Concentric Circles

Circles with the same center are concentric circles.







Real World Connection

A car has a turning radius of 16.1 ft. The distance between the two front tires is 4.7 ft. In completing the outer turning circle, how much farther does a tire travel than a tire on the concentric inner circle?







Arc Length

Arc length is a fraction of the circles circumference.







Theorem 10-10:“Arc Length”

The length of an arc of a circle is the product of the ratio of the arc over 360º and the circumference of the circle.







Finding Arc Length

Find the length of arc PQ.

42

120

P

Q







Congruent Arcs

Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles.







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Chapter 10: Area