Discover & Approximate Pi with Circles and Regular Polygons
1. Discovery of the number Pi
2. Objective: Students will use their existing knowledge on how to find the perimeter of a regular polygon to approximate the number Pi using the relationship between an increasing n-sided regular polygon as n approaches infinity. This relationship will correspond with the common core standards found on page 77 of http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf where it is stated under “Circles” to understand and apply theorems about circles. Students will be using knowledge of measurements of polygons supported by the common core standards found at the bottom of page 25 of the same link document.
Mini-Experiment Task Sheet:
Please recall what we have learned about areas and perimeter of a regular polygon (RP).
Area of RP=(1/2)(altitude)(perimeter) and Perimeter of RP=(number of sides)(length of one side)
Go to http://www.geogebratube.org/student/m11596 and record your findings on the following prompts using this Geogebra Link (show all of your algebra work and thought process):
Please note that you can manipulate the number of sides of this shape by using the slider tool where you can have a shape of no less than 3 and no more than 100 sides. Also, please note that Every shape is a regular polygon and that the sides will remain constant. The Altitude (FD) of the polygon and radius (FA) of the circle will change depending on the number of sides you select. The lengths are marked so that you can use that information to calculate your responses. Also please note that you may need to zoom (Hold Ctrn button and hit “+” or “-“ or the Command button on a Mac) out to see the shape as your sides increase. Use the white arrow at the Every polygon will be inscribed inside the circle.
1. What is the area and perimeter of this triangle?
2. Choose a new polygon with sides no more than 10 but greater than 3 and find the area and perimeter. (please note how many sides you choose).
3. Repeat prompt 2 but choose a regular polygon with sides greater than 35 but less than 75.
4. Now manipulate the slide bar, what shape do you see this polygon forming into as the number of sides increases? (please write your answer in 1-3 complete sentences)
5. Calculate the area and perimeter of one more polygon with sides greater than or equal to 75.
6. (Note that the radius of the circle is always going the vertex A which is always on the circumference of the circle.) With that in mind do the following calculations from each of your results from prompt 1, 2, 3 and 5: calculate perimeter/(2)(radius) and calculate area/(radius)^2 and write your number in decimal form below.
7. Write 3-4 sentences on the observations you have made about the results you and your class mates got.
8. As a class come up with a hypothesis as to why your numbers appear to be so close.
9. What more could you do to test your hypothesis?