Expert: David Hemmer - 4/19/2007

Question
hi sir, im college student and i got question and expected will come out in exam, thank you sir.. the question is:

Archimedes and other estimated the ratio of circumference of a circle to its diameter (ð) by using regular polygons – the greater number of sides of the polygons, the more accurate the estimate ð.

(a)   Consider polygons inscribed in a circle of radius 1. Start with a hexagon inscribed in a circle of radius 1, find the perimeter of the hexagon. Find the approximation of 2ð. (Hint: the hexagon consist of six equilateral triangles)
(b)   Find the perimeter of 12-gons, and estimate the value of 2ð.
(c)   Find the perimeter of 24-gons, and estimate the value of 2ð.
(d)   Find a formula for the length y of the side of a regular 2n-gon in terms of the length x of the side of a regular n-gon.
(e)   Discuss the value of ð.


Answer
This will be very hard for me to answer without drawing pictures. I suggest you visit the website:

http://www.pbs.org/wgbh/nova/archimedes/pi.html

this describes exactly the method you are using to approximate Pi. Write me back if you don't understand anything.