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2. A sector has radius 8 km and a central angle of 0.5 radians. What is the length of the arc enclosed by the angle?
3. One sector has a radius of 10 meters and a central angle of radians. A sector of a
di erent circle has a radius of 7 meters. What central angle would the second sector need in order to trace out the same arc length as the rst sector?
4. One sector has a of radius 15 inches and a central angle of 3 radians. A sector of a di erent circle has a central angle of 0.2 radians. What does the radius of the second sector need to be in order for the arc lengths to be the same?
Student Response
1. cm, or approximately 3.14 cm
2. 4 km
3. radians, or approximately 0.30 radians 4. 225 in
Activity Synthesis
Ask previously identi ed students to share their responses to question 2. This would be a good opportunity to remind students of the de nition of radian measure as the ratio of arc length to radius. Using this de nition can sometimes be an easier way to reason about arc length.
Lesson Synthesis
In this lesson, students practiced solving problems involving arc length and sector area. Here are some questions for discussion:
• “How are solving problems involving circular arc length the same as problems involving sector area? How are they di erent?” (Problems involving arc length and sector areas are similar because they both involve reasoning about parts of a whole circle. In the case of arc length, it is often helpful to reason about parts of a whole circumference, whereas sector area is concerned with parts of the area of the whole circle. For arc length, it is sometimes easier to use the de nition of radian measure to express the connection between angle, arc length, and radius.)
• “Circular arc length can describe the crust of a slice of pizza, and sector area can describe the slice itself. What other examples in the world can you think of that involves arc length or sector area?” (For example, the area wiped by a windshield wiper is roughly a sector area. Arc length can describe the length of the path of an object in circular motion, like Earth’s orbit around the sun or the path of a person traversing across the spherical earth itself.)
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Teacher Guide