Topic: Area and Arclength of a Circle

Learning Objectives:

  • The Student Will apply formulas for area and perimeter of a circle
Prior Knowledge:. Geometry of a circle, areas of regular polygons.

Motivational Problem: Have the students find the area of several regular n-sided polygons inscribed in a circle of radius 1. If the students know trigonometry, they should be able to do this without assistance; otherwise provide them with the appropriate apothem lengths.

Overview

Have students find areas for several values of n: say, 4, 5, 6, 7, 8. Have them try a very large value--say, n=50, and then n=1000. Ask them to draw this polygon--it is virtually indistinguishable from a circle, and its area (when rounded) is approximately 3.14. Note that this is *not *a proof that the area of a circle is pi times the radius squared, just an investigation. A proof would have to come much later, requiring much more advanced mathematics.

For sectors and arc-length, the key is to remember that in each case, we are merely looking at some portion of the full circle. A proportion will always be used to compute sector area or arc-length--the question to be asked is always "What percentage of the full circle am I using?" The answer to that question, along with the area or circumference of the original circle, will be sufficient to find sector area and arc-length.

Key Questions: How can we approximate the area of a circle? How can we approximate the perimeter of a circle? What is the strategy for finding arc-length and sector area?

Looking Ahead: Arc length and sector area are important ideas in trigonometry, especially when it comes to the concept of radian measure for trigonometric functions.

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