Estimate Pi using perimeter:
- If a circle has circumference (note this is just a fancy way of saying perimeter) pi, what would its radius be? ______
Suppose regular n-sided polygons are inscribed and circumscribed using this circle.
Inscribed
Regular polygon: a regular polygon is______
Start with a 3 sided regular figure.
What is the measure of angles 1 and 2?______
Why?______
If a figure has n sides, what is each of the angles? ______
Find the perimeter of the triangle.
How will this value relate to PI? How can we get a better estimate?
Pick a number of sides between 5 and 18. ______Find the perimeter of this polygon inscribed in the same circle above.
Determine a formula where the value of n (the number of sides) will produce the perimeter of the regular polygon.
Now try circumscribed perimeter. How will this value relate to PI?
Find the formula for the perimeter of an n-sided regular polygon inscribed in the circle above.
AREA: Using the same principles, now think in terms of area.
(Note the circle must change to make the area of the circle equal to PI)
Final Product
Show where the formulas come from (diagrams and explanations)
Solve upper and lower bound for n between 5 and 15 (show work)
2 formulas (one for upper bound and a second for lower bound)
Graph for each formula( may want to also graph y=π)
Explanation of upper and lower bound and graph
For perimeter determine the number of sides necessary for a regular polygon such that if I inscribe a circle and circumscribe a circle in the same polygon, I get pi accurate to 4 decimal places. (3.1415) (10 pts)
History/Usefulness/Interesting
Search the web and provide a brief history of the ancient groups that used PI, why they needed it, and how accurate their approximations were.—Provide a bibliography of sources (10pts)
Area: same problems as above (Extra Credit) A= π r2 (10 pts)
GRADE: (40 points) There will be an in class writing assignment based upon the work above. Student will be able to examine their formulas, graphs, and images.
Dr. Kevin A Thompson