10-6

Circles and Arcs

Learning Goals

1. To find the measures of central angles and arcs.

2. To calculate the circumference and arc length.

What is the measure of the angle all the way around B?

B C

What is the measure of the angle all the way around B?

360˚

B C

If m ABC = 60°,

B C

A

If m ABC = 60°,

then the m ABC = ?

B C

A

If m ABC = 60°,

then the m ABC = 300°.

B C

A

Central Angle

An angle whose vertex is at the center of a circle.

CentralAngle

Arcs

The sections of the circle separated by the central angles.

arcs

Arcs are measured by their corresponding central angle.

m ABC = 80

m AC = ?

A

B

CD

80°

Arcs are measured by their corresponding central angle.

m ABC = 80

m AC = 80

A

B

CD

80°

Arcs are measured by their corresponding central angle.

m ABC = 80

m AC = 80

m ADC = ?

A

B

CD

80°

Arcs are measured by their corresponding central angle.

m ABC = 80

m AC = 80

m ADC = 280

A

B

CD

80°

Minor arc: less than 180° and 2 letters are used for the name.

Major arc: greater than 180° and 3 letters are used for the name.

***Important***

When writing arc measure an “m” must be used.

Example:

m AC = 80°

Semicircle

An arc whose measure equals 180.

Circumference

C = 2πr

Arc Length

The measure of the length of an arc.

***Important***

When writing arc length no “m” is used.

Example:

PB = 3π

Calculating Arc Length

Arc Measure

360• Circumference

Calculating Arc Length

(which is the same as)

Arc Measure

360• Circumference

Calculating Arc Length

(which is the same as)

Arc Measure

360• Circumference

Arc Measure

360• 2 π r

Calculating Arc Length

1) Divide the arc measure by 360.

P

BJ90°

903604m

Calculating Arc Length

2) Find the circumference.

P

BJ90°

903604m

C = 2πrC = ?

Calculating Arc Length

2) Find the circumference.

P

BJ90°

903604m

C = 2πrC = 8π

Calculating Arc Length

3) Multiply the two.

P

BJ90°

903604m

C = 2πrC = 8π

90360

=• 8π 2π

Calculating Arc Length

PB =

P

BJ90°

4m

2π

Is there a maximum arc length?

Is there a maximum arc length?

No, it is a distance.

Is there a maximum arc measure?

Is there a maximum arc measure?

Yes, 360.

Turn to page 654.

9, 10, 33, 3411-18, 24-26, 29-32, 39-43, 48

5 Points: 100% Complete 4 Points: 80% Complete 3 Points: 60% Complete 2 Points: 40% Complete

1 Point: 20% Complete