Chapter 10 Area Section 10 6 Circles and

Chapter 10: Area Section 10 -6: Circles and Arcs

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.

Example To 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:

Circumference The circumference is the distance around the circle.

Pi p The 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 diameter. times the

Concentric 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. P Q

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