Lesson 2 From Polygons to Circles Solidify Understanding

Jump Start

1.

Represent each of the following situations with a table, a graph, and an equation:

a.

Your family is planning a trip, and you have stopped at the gas station to fill up the car. You notice that the gas pump dispenses gas at a rate of $2.5$ gallons per minute.

b.

Once you are on the road you set the cruise control at $75$ miles per hour.

c.

You notice that at this speed, your car’s fuel efficiency is $30$ miles per gallon.

2.

What do all of these problems have in common?

Learning Focus

Relate the circumference and area of circles to the perimeter and area of regular polygons.

Where did $π$ and the formulas for calculating the circumference and area of a circle come from?

Open Up the Math: Launch, Explore, Discuss

Part 1: From Perimeter to Circumference

In the previous lesson, Planning the Gazebo, you developed a strategy for finding the perimeter of a regular polygon with $n$ sides inscribed in a circle of radius $r$ . Tehani’s strategy consists of the following formula:

$P = 2 n \cdot r \sin (\frac{360 °}{2 n})$ Tehani drew this diagram as part of her work developing this formula.

1.

Using Tehani’s diagram, explain how she arrived at her formula.

2.

Since $n$ is the only thing that varies in this formula, Travis suggests that Tehani might rewrite her formula in the form

$P = 2 r [n \cdot \sin (\frac{360 °}{2 n})]$ . Because the perimeter of an $n$ -gon approximates the circumference of a circle, when $n$ is a large number of sides, Travis suggests they examine what happens to the $n \cdot \sin (\frac{360 °}{2 n})$ portion of Tehani’s formula as $n$ gets larger and larger. Use a calculator or spreadsheet to complete the following table to see what happens.

$n$	$n \cdot \sin (\frac{360 °}{2 n})$
$6$
$12$
$24$
$48$
$96$
$100$
$1, 000$
$10, 000$

3.

As the number of sides $n$ of the polygon increases, what happens to the value of the expression $n \cdot \sin (\frac{360 °}{2 n})$ ?

4.

Write a formula for the circumference of a circle based on Tehani’s formula for the perimeter of an inscribed regular $n$ -gon and what you have observed while generating this table.

Part 2: From the Area of a Polygon to the Area of a Circle

Area: approach #1

Tehani’s formula for the area of a regular polygon with $n$ sides inscribed in a circle of radius $r$ is:

$A = \frac{1}{2} \cdot 2 n \cdot r \sin (\frac{360 °}{2 n}) \cdot r \cos (\frac{360 °}{2 n})$

5.

Explain in detail how Tehani arrived at this formula. You may refer to Tehani’s diagram in problem 1.

6.

Travis suggests that they might rewrite Tehani’s formula in the form $A = r^{2} [n \cdot \sin (\frac{360 °}{2 n}) \cdot \cos (\frac{360 °}{2 n})]$ and then examine what happens to the last part of the formula as $n$ gets larger and larger. Use a calculator or spreadsheet to complete the following table, and see what happens.

$n$	$n \cdot \sin (\frac{360 °}{2 n}) \cdot \cos (\frac{360 °}{2 n})$
$6$
$12$
$24$
$48$
$96$
$100$
$1, 000$
$10, 000$

7.

As the number of sides $n$ of the polygon increases, what happens to the value of the expression $n \cdot \sin (\frac{360 °}{2 n}) \cdot \cos (\frac{360 °}{2 n})$ ?

8.

Write a formula for the area of a circle based on Tehani’s formula for the area of an inscribed regular $n$ -gon and what you have observed while generating this table.

Area: Approach #2

A circle can be decomposed into a set of thin, concentric rings, as shown on the left in the following diagram. If we unroll and stack these rings, we can approximate a triangle as shown in the figure below.

9.

Describe the height of this “triangle” relative to the circle.

10.

Describe the length of the base of this “triangle” relative to the circle.

11.

As the rings get narrower and narrower, the triangular shape gets closer and closer to an exact triangle with the same area as the circle. What would this diagram suggest for the formula of the area of a circle?

Area: Approach #3

A circle can be decomposed into a set of congruent sectors, as shown in the following diagram. We can rearrange these sectors to approximate a parallelogram as shown.

12.

Describe the height of the “parallelogram” relative to the circle.

13.

Describe the base of this “parallelogram” relative to the circle.

14.

As we decompose the circle into more and more sectors, the “parallelogram” shape gets closer and closer to an exact parallelogram with the same area as the circle. What would this diagram suggest for the formula for the area of a circle?

Ready for More?

1.

a.

Complete the following table for the area of various circles using the formula derived in the exploration: $A = π r^{2}$ . Approximate the area to the nearest whole number by using $π \approx 3$ .

$r$	$r^{2}$	$A$
$0$
$1$
$2$
$3$
$4$
$5$

b.

Plot $A$ as a function of $r$ .

c.

Plot $A$ as a function of $r^{2}$ .

2.

Is the area of a circle proportional to its radius? Why or why not?

3.

Why is it appropriate to refer to $π$ as a constant of proportionality for area? What is the area of a circle proportional to?

Takeaways

The perimeter of a regular polygon is proportional to the diameter of the circle in which it can be inscribed.

The constant of proportionality $k$ is different for each different type of regular polygon (hexagon, octagon, dodecagon, etc.), but can be calculated using the formula

As the number of sides of the regular polygon inscribed within a circle increases, the perimeter of the polygon

This leads to the formula for the circumference of a circle:

Using similar reasoning, as the number of sides of the regular polygon inscribed within a circle increases, the area of the polygon

This leads to the formula for the area of a circle:

Vocabulary

convergence
limit (convergence)
Bold terms are new in this lesson.

Lesson Summary

Previously, we have been given formulas for the circumference and area of a circle and have used them in application problems. In this lesson, we learned where these formulas came from and gained a better sense of why $π$ is such an important number, and also how it occurs naturally in the context of relating regular polygons to the circles in which they are inscribed.

Retrieval

1.

Find the perimeter and area of the figure.

(Assume the diameters of the cone and the ice cream are the same.)

2.

Use the given circle to find the indicated measures.

a.

circumference of the entire circle

b.

the fractional part of the circle

c.

the shortest arc length $\overset{⌢}{B C}$ that is intercepted by the central angle

(The shortest arc length from $B$ to $C$ is called the minor arc. The long distance from $B$ to $C$ around the circle is called the major arc.)