Lesson 6 Pied! Solidify Understanding

Jump Start

The words sector and arc length have been used in previous lessons, but since they are the focus of today’s lesson, let’s review their meanings.

Highlight and label a sector and an arc length on the following diagram.

Wendell has shaded a portion of the diagram, and wants to know if this region has a name. His teacher told him that it is referred to as a segment of a circle.

Learning Focus

Find formulas for arc length and area of a sector of a circle.

We all like pie, but how can you figure out the portion of the total pie you are eating when you take the biggest slice?

Open Up the Math: Launch, Explore, Discuss

Students have planned several activities to celebrate Pi Day at their school. In addition to pie eating contests and “pie-ing” their favorite teachers, the Math Club plans to make money by selling slices of pie during lunch hour. Each member of the club has contributed a couple of homemade pies for the sale. Unfortunately, the members chose a variety of sizes and shapes of pans to bake their pies in. Some students used $9 –inch$ round pans for their pies, others used $8 –inch$ round pans, a few used $8 \times 8 inch$ square pans, and one student used a $9 \times 13 inch$ cake pan for his pie. Now the club members have the dilemma of how to slice the pies so each slice is about the same amount, since they plan to charge the same amount for each slice of pie, regardless of the pan it came from.

After much debate, the club members have decided to slice the $8 -inch$ round pies into $5$ equal slices (or sectors, as the math club members call them), the $9 -inch$ round pies into $6$ equal slices, the $8 \times 8 inch$ pies into $2 \times 4 inch$ rectangles, and the $9 \times 13 inch$ pie into $3 \times 3 \frac{1}{4} inch$ rectangles. Although the pieces look like they are all about the same size, some students think there might be a price advantage in buying one type of slice over another.

1.

Which slice of pie is the largest, and which is the smallest? How did you decide?

Unfortunately, not everyone in the Math Club is good at eye-balling equal size sectors when cutting round pies. Therefore, one of the students is assigned to be in charge of “quality control.” He is given a protractor and is told to reject any slices of pie that are more or less that $4 °$ from the exact angle measurement.

2.

Using this criterion, what is the smallest and largest area of pie you might get in a slice of pie taken from the $8 -inch$ pan?

3.

Using this criterion, what is the smallest and largest area of pie you might get in a slice of pie taken from the $9 -inch$ pan?

The student in charge of quality control finds it is too difficult to measure the angle of a sector of pie in degrees and suggests that they cut a piece of string that can be used to measure around the outer edge of the pie to let the servers know where to make the next cut.

4.

How long should this string be to measure the arc of a slice of pie for the $8 -inch$ round pies?

5.

How long should this string be to measure the arc of a slice of pie for the $9 -inch$ round pies?

Pause and Reflect

Wendell really likes pie and has offered to pay double the price for a slice of pie that is guaranteed to contain at least $15 {in}^{2}$ of pie.

6.

What is the degree measure of the smallest sector of the $8 -inch$ round pie that will satisfy Wendell’s cravings?

7.

How long should the string be to measure the outer arc of this sector?

8.

What is the degree measure of the smallest sector of the $9 -inch$ round pie that will satisfy Wendell’s cravings?

9.

How long should the string be to measure the outer arc of this sector?

10.

A sector of the $9 -inch$ round pie measures $n °$ . What is its area? What is its arc length?

Ready for More?

Wendell has noticed that many people throw away much of the pie crust—his favorite part of the pie—and wants to sell pies shaped like regular polygons by cutting off the segments and eating them. He wants to know how much of the pie is removed by cutting off each segment.

Develop a formula that Wendell can use to find the area of a segment of a pie depending upon the radius of the pie and the type of $n$ -gon that will be created by cutting off the segments.

Takeaways

The formulas for the arc length and area of a sector for a circle of radius $r$ and a central angle of $n °$ are:

Arc length =

Area of sector =

Adding Notation, Vocabulary, and Conventions

Define each of the following parts of a circle:

A sector of a circle is

An arc length is

A segment of a circle is

Vocabulary

arc length
sector
segment of a circle
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we found a relationship between arc length and the area of a sector of a circle bounded by the arc and the two radii of the circle drawn to the endpoints of the arc.

Retrieval

1.

Find the ratios.

$\sin A =$	$\sin B =$
$\cos A =$	$\cos B =$
$\tan A =$	$\tan B =$

2.

Find the measure of angle A.