# Overview

## Introduction to Circles

Developed by Paul Hartzer, Spring 2012
Purpose: This is the first lesson in a unit on the parts and relationships within a circle. The basic textbook is Glencoe’s Geometry, Michigan Edition.

# Objectives and Materials

## Objectives

Identify the pattern of the perimeters of polygons of increasing numbers of sides.
Get a basic understanding of limits.
Learn the formula for the circumference of a circle.

## HCSEs

L1.1.6 Explain the importance of the irrational numbers and in basic right triangle trigonometry, and the importance of pi because of its role in circle relationships.
L2.2.3 Use iterative processes in such examples as computing compound interest or applying approximation procedures.

## Resources and Materials Needed

Microsoft Excel and GeoGebra (for individual use or demonstration)
Basic or graphing calculators, one for each student

# Lesson

## Introduction and Lesson

Begin with the handout. Guide students minimally. The purpose of this exercise is for them to notice that the number in the fourth column is converging on pi. Then discuss the importance of circles in general, tying the topic to circular objects in their life (wheels, cans), and prepare them for learning new vocabulary.
Once the students have been given time to explore the worksheet alone or in groups, bring them together to discuss their findings. Refer to the “Polygon half perimeter” Excel file (“Inscribed” tab), as well as to the “Inscribed Polygons” GeoGebra file. These demonstrate how, as polygons have increasingly high numbers of sides, they get closer to resembling circles. If possible, allow the students to explore the GeoGebra file on their own.
In the GeoGebra file, the square and hexagon have been marked to allow for a discussion of trigonometry. Students will likely have the most difficulty with the specific formula being used; the formula for circumscribed polygons (day 5) is simpler, but relies on the notion of tangents. Also, the inscribed polygon formula is used first because it more closely represents the historic development.

## Practice and Closure

The practice comes in the form of the constructive worksheet.
Review the lesson. Point out that they’ve now learned an important building block of calculus.