Unit 7, Lesson 10
Angles, Arcs, and Radii
Geometry
10.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students continue to explore the relationship between arc length, radius, and central angle. They recognize that a particular central angle defines arc lengths of different sizes in circles with different radii.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Activity
NoneHan and Tyler are each completing the same set of tasks on an online homework site. Han is using his smartphone, and Tyler is using his tablet. Each student’s progress is indicated by the circular bar shown in an image. The shaded arc represents the tasks that have been completed. Once a student has finished all the tasks, the full circumference of the circle will be shaded.
Han's progress
Tyler's progress
Tyler says, “The arc length on my progress bar measures 4.75 centimeters. The arc length on Han’s progress bar measures 2.25 centimeters. So I’ve completed more tasks than Han has.”
- Do you agree with Tyler? Why or why not?
- What information would you need to make a completely accurate comparison between the two students’ progress?
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
The goal of the discussion is to start to build the idea that the length of an arc and the radius of the circle give us information about the central angle that defines the arc. Ask students these questions:
- “There are actually two circles that make up each progress bar, an inner circle and an outer circle. Which one should we focus on when measuring progress?” (It doesn’t matter; both contain the same information.)
- “If the length of the arc on Tyler’s circle is longer, why does that not necessarily mean he has completed more tasks?” (The completed proportion depends not only on the arc length but also on the size of a circle.)
- “What information do you think would be helpful to accurately compare the progress of the two students?” (Students may discuss proportions and fractions. If the phrase “central angle” does not come up, ask students what information this measurement would give us.)
- “What measurement can we use to compare the ‘sizes’ of the circle?” (Radius gives relative size.)
Student Lesson Summary
If we have the same central angle in two different circles, the length of the arc defined by the angle depends on the size of the circle. So, we can use the relationship between the arc length and the circle’s radius to get some information about the size of the central angle.
For example, suppose Circle A has radius 9 units and a central angle that defines an arc with length . Circle B has radius 15 units and a central angle that defines an arc with length . How do the two angles compare?
For the angle in Circle A, the ratio of the arc length to the radius is , which can be rewritten as . For the angle in Circle B, the ratio of the arc length to radius is , which can also be written as . That seems to indicate that the angles are the same size. Let’s check.
Circle A’s circumference is units. The arc length is of , so the angle measurement must be of 360 degrees, or 60 degrees. Circle B’s circumference is units. The arc length is of , so this angle also measures of 360 degrees, or 60 degrees. The two angles are indeed congruent.