Lesson 2: Area of a Circle

Introduction

Student Audience:

The target audience is students studying Euclidean geometry.  As written, this lesson is anticipated to fill 1.5 one-hour class period.

Objectives:

¤       Students will use previously studied formulas for the area of a regular triangle, quadrilateral, and hexagon to compute the area of each of these figures when circumscribed around a particular circle

¤       Students will develop a general formula for the area of a regular polygon using right triangle trigonometry

¤       Students will compare the area of a polygon to the area of its circumscribed circle

¤       Students will graph, with the aid of technology, the area of circumscribed polygons for a given circle

¤       Students will analyze the discrete area graphs to develop hypotheses about the value of the area of a particular circle

¤       Students will gather data about the radius and estimated area for various circles and create a graph displaying this data

¤       Students will analyze radius and estimated area data, looking for patterns and relationships

¤       Students will develop a formula for calculating the circumference of any circle given the radius of that circle

Mathematical Concepts:

¤       Area of a regular polygon

¤       Area of a circle

¤       Right triangle trigonometry (including sine, cosine, and tangent functions)

¤       Graphing functions

¤       Asymptotes of a graph

¤       Calculating angles of a regular polygon

Lesson Synopsis:

Students will begin by using prior knowledge to calculate the area of familiar regular polygons that have been circumscribed around a given circle.  Students will then develop general formulas for finding the area of any regular polygon circumscribed around a circle with a given radius. Using spreadsheet software, students will create a table of values of the area of a regular n-gon circumscribed around a particular circle for various values of n.   By graphing these data, students will generate hypothesis about the area of the specific circle.  Students will also use the spreadsheet to estimate the area of circles having various radii lengths.  This data will be graphed and analyzed, leading to a formula to compute the area of any circle.

Lesson

Materials:

¤       Classroom computer projector

¤       GeometerÕs Sketchpad software or similar

¤       Circle Area worksheet

Implementation:

The instructor will begin by posing the following question:  Will the area of a polygon that is circumscribed around a circle be larger than, smaller than, or equal to the area of the circle?  Students should also form hypotheses comparing the area of a circle to the area of any inscribed polygon. Using software such as GeometerÕs Sketchpad, the instructor may choose to circumscribe a circle with various regular polygons (e.g. equilateral triangle, square, pentagon, hexagon, octagon) facilitate a discussion of which polygon seems to have area closest to the circle.

Using the Circle Area worksheet as a guide, students will work individually or in pairs to develop a general formula for finding the area of a circumscribed polygon in terms of the radius of the circle and the number of sides of the regular polygon.  The following formula is likely to be developed:

Area of Circumscribed Polygon

Next, students will utilize spreadsheet software to generate a table of values of the area of many different regular polygons circumscribed around a given circle.  A sample spreadsheet can be found here.  After examining the relationship between the number of sides of the polygon, the area of the polygon, and the area of the circle, students will create a second spreadsheet comparing the circle radius to approximate circle area (as determined by calculating the area of a regular circumscribed n-gon where n is very large).  Students will then chart their data set and make conjectures about the relationship between a circleÕs radius and area. A sample data set and chart can be found here.

Facilitator Questions:

¤       How does the area of the polygon change as the number of sides increases?

¤       How does the area of the circumscribed polygon compare to the area of the circle as the number of sides is increased?

¤       How does the estimated circumference of a circle change as the radius of the circle increases?

¤       Does the relationship between the radius of a circle and the estimated area appear linear? Quadratic? Exponential? Something else?

¤       Using your sample data points, what equation would you guess would the linear relationship that we see?

Assessment

Students will be assessed informally during the class activity as the instructor circulates the classroom, asking questions to guide students in their exploration.  Additionally, student work will be assessed for completion, correctness of computation and general formula, and quality of spreadsheet creation, data collection, and data analysis.  In particular, the instructor will evaluate each studentÕs ability to gather, organize, and make conjectures from spreadsheet data.

NAME __________________________________               DATE _____________

Circle Area

During the following investigation of circles you will look at the area of lots of polygons and try to make estimates about the area of circles.  You will also be trying to figure out if there is any relationship between the radius of a circle and its area.  Be sure to organize your work and be prepared to present your findings to the class.

Take a look at the figure below.  In this diagram a regular triangle (er . . . equilateral triangle) has been circumscribed around a circle with radius r.

Your first task is to find the area of the triangle in terms of the radius, r, of the circle.

Triangles are not the only regular polygons that can be circumscribed around our circle.  In fact, any regular polygon can be circumscribed around this circle.  Your second task is to figure out a formula for the area of a regular n-gon (polygon with n sides) that has been circumscribed around our circle.  This formula will depend on r (the radius of the circle) and n (the number of side of the polygons).  You may want to use the unfinished sketch of a circumscribed n-gon below to help you.

Once youÕve figured out a candidate for your general formula, your third task is to check this formula by using it to compute the area of a circumscribed equilateral triangle Compare the value to the area you calculated in your first task.

Additionally, sketch a picture of our circle with a square circumscribed around it.  According to your sketch, what is the area of the square in terms of r?  Does that match the value you compute using your candidate formula?

Using spreadsheet software, your fourth task is to create a table of values of the areas of various n-gons that have been circumscribed around a circle with radius equal to 1.  How does the area of the polygon change as the number of sides gets larger and larger?  How does the area of the polygon compare to the area of the circle as n gets very large?  How might you use your spreadsheet to estimate the area of a circle with radius 1?  With radius 4?

Using the table below and your spreadsheet, your fifth task is to gather and organize data about the radius and estimated circumference of lots of circles.  After you have collected your data, create a graph of the data (you can use spreadsheet software or create a graph by hand).

Make conjectures about the relationship between the radius of a circle and its area.

 Radius of Circle Estimated Area of Circle