Jump Start

In a previous Exit Ticket, you examined this diagram, in which Tehani drew some similar figures, based on the fact that all circles are similar and that circle was the image of circle following a dilation. Tehani found that the dilation centered at point had a scale factor of .

Here is another calculation Tehani made:

• She calculated the length of the intercepted arc and found it to be .

Based on this information, find the length of the intercepted arc .

Learning Focus

Develop a new unit for measuring angles.

We are very familiar with measuring angles in degrees. But why are there in a circle? Why not , or ?

Are there other ways of measuring angles that might be more natural to the features of the circle and maybe wouldn’t require additional tools like a protractor?

Open Up the Math: Launch, Explore, Discuss

Last year Madison won the city’s “Most Outstanding Garden” Award for her square garden. This year she plans to top that with her design for a beautiful round garden. Madison’s design starts with a sprinkler in the center and concentric rings of colorful flowers surrounding the central sprinkler. Pavers will create both circular pathways and pathways that look like spokes on a wheel between the flowers. The sprinkler can be adjusted so it waters just the inner circle of flowers, or it can be adjusted so it waters the entire round garden. Consequently, flowers that need to be watered more frequently will be placed near the center of the garden, and those that need the least amount of water will be placed farthest from the center. The sectors of the garden will not all be the same size, since they need to accommodate different types of plants.

Here is Madison’s design for her garden. The number of degrees in each sector has been marked.

1.

Madison has marked the degree measure on the arcs of the outermost ring of the garden. Determine the angle measures for the arcs on the inner and middle rings of the garden.

2.

Madison needs to order pavers for the garden. She plans to vary the size and color of the pavers in different parts of the garden. Consequently, she needs to know the lengths of different portions of the paths. Help her complete this table by calculating the missing arc lengths.

 Distance from Center Arc Length $40°$ Sector $50°$ Sector $60°$ Sector $90°$ Sector $120°$ Sector Inner Circle of Pavers Middle Circle of Pavers Outer Circle of Pavers

3.

As Madison fills out the table, she begins to notice some important relationships as she examines the sequence of numbers in each column of the table. What do you notice?

4.

One thing Madison notices involves the ratio of the arc length to the radius of the circle. Complete this version of the table, and state what you think Madison notices.

 Distance from Center Arc Length / Radius $40°$ Sector $50°$ Sector $60°$ Sector $90°$ Sector $120°$ Sector Inner Circle of Pavers Middle Circle of Pavers Outer Circle of Pavers

As Madison examines these numbers, she realizes that all arcs in the same sector have the same degree measurement, and all arcs in the same sector have the same value for the ratio of arc length to radius. This makes her wonder if these new numbers could be used as a way of measuring angles, just as degrees are used.

Later in the evening, Madison shares her discovery with her older sister Katelyn, who is taking calculus at a local university. Katelyn tells Madison that her new numbers for measuring angles in terms of the ratio of the arc length to the radius are known as radians and that they make the rules of calculus much easier than if angles are measured in degrees.

Madison learns so much from examining the arc length of the sectors of her garden that she decides to examine the areas of the sectors also.

Complete this table for Madison by calculating the areas of the sectors for the different rings of the garden.

 Distance from Center Area of Sector $40°$ Sector $50°$ Sector $60°$ Sector $90°$ Sector $120°$ Sector Inner Circle of Pavers Middle Circle of Pavers Outer Circle of Pavers Extended Circle of Pavers

What patterns do you notice in the way the numbers in each column of this table increase?

Takeaways

Using precise language, define or describe what a radian is:

Vocabulary

• Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about a new unit for measuring angles called a radian. We developed this new unit of angle measurement by examining the ratio of arc length to radius for various central angles and for various distances from the center of the circle.

Retrieval

1.

Find the volume and the surface area of the cylinder in Figure 1.

2.

a.

Calculate the length of and and the areas of each sector.

Find the ratios:

, , , and .