10.6/10.7 Circles, Arcs, Segments, and Sectors Objectives: Find the Measures of Central Angles and Arcs Find Circumference and Arc Length Find Areas of Circles Find Areas of Sectors and Segments of Circles

What is a Circle? A Circle is the set of all points equidistant from a given point called the Center. You can name a Circle by its Center. Circle P (P) is shown at the Right.

Other Circle Terminology… A Radius is a segment that has one endpoint at the center and one endpoint on the circle. Segments PC, PA and PB are Radii. Congruent Circles have Congruent Radii. A Diameter is a segment that has endpoints on the circle and contains the center of the circle.

What is a Central Angle? The Central Angle is an angle whose vertex is the Center of the Circle. <CPA is a Central Angle. Remember there is 360° in a Circle.

What is an Arc? An Arc is part of a Circle. A Semicircle is half of the Circle. A Minor Arc is smaller then a Semicircle. A Major Arc is greater then a Semicircle.

Arcs… To name a Minor Arc use only the two endpoints of the arc. Minor Arc Measure is 100° Major Arc Measure is 360° – 100° =260° Arc Measure is 180° 180° 100° 100° To name a Minor Arc use only the two endpoints of the arc. To name a Major Arc or Semicircle, use three points with the endpoints at the beginning and end.

Your Turn: XP Battle

Arc Measure The measure of a minor arc is defined as the measure of its central angle: The measure of a major arc is 360o minus the measure of the corresponding minor arc:

The following information was determined from a survey determining how people really spend their time. 7% 31% 18% 20% 15% 9% Directions: Find the measure of each central angle to the nearest whole number. Find the corresponding percent of 360. 1.) Sleep 4.) Work 2.) Other 5.) Entertainment 3.) Food 6.) Must Do

Arc Addition Postulate Adjacent arcs are arcs on the same circle that have exactly one point in common. You can Add the Measure of Adjacent Arcs just as you can add the measures of adjacent angles.

Your Turn: XP Battle

circumference of a circle – the ___________ around the circle. distance or r d Circles that lie in the same plane and have the same center are ___________________. concentric circles

A car has a turning radius of 16. 1 feet A car has a turning radius of 16.1 feet. 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? To find the radius of the inner circle, subtract 4.7 ft from the turning radius. circumference of outer circle: radius of the inner circle: circumference of inner circle: The difference of the two distances: A tire on the turning circle travels about 29.5 ft farther than a tire on the inner circle.

Arc Length The measure of an arc length is a fraction of the circles circumference. The Fraction is based on a ratio of the measure of the central angle out of 360°

Your Turn: XP Battle

Try Some More! Find the Arc Measure of AB and CD. Find the Arc Length of AB and CD What do you notice about the Arc Measure? The Arc Length?

Be Careful… Two arcs can have the same Arc Measure but different Arc Lengths. It is also possible for two arcs to have different Arc Measures but the same Arc Lengths. Congruent Arcs have the same Arc Measure and Same Arc Length.

Area of a Circle Area of a Circle: The area of a circle is the product of π and the square of the radius. A = πr2 What is the area of a circle with diameter 16?

Sectors A sector of a circle is a region bounded by an arc and the two radii to the arc’s endpoints. You name a sector using one arc endpoint, the center of the circle, and the other arc endpoint. Area of a Sector: The area of a sector of a circle is the product of the ratio of the measure of the arc and 360˚ and the area of the circle.

Finding Areas of Sectors What is the area of sector GPH? Leave your answer in term of π. 𝑚 𝑎𝑟𝑐 𝐺𝐻 360° 𝑥 𝜋𝑟2 72° 360° 𝑥 𝜋 15 2 45𝜋

Your Turn: XP Battle A circle has a radius of 4 in. What is the area of a sector bounded by a 45˚ arc? Leave your answer in terms of π. 2𝜋

Segments of Circles A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle. To find the area of a segment, draw radii to form a sector. 𝑚 𝑎𝑟𝑐 𝐴𝐵 360° 𝑥 𝜋𝑟2 − 1 2 𝑎 𝑏 𝑠𝑖𝑛𝐶=𝐴

Finding the Area of a Segment of a Circle What is the area of the shaded segment? Round your answer to the nearest tenth.

Your Turn: XP Battle What is the area of the shaded segment? Round your answer to the nearest tenth. 4.6m2