Expert: Steve Holleran - 4/19/2007

Question
hai sir, can you solve this question for me? i expected it will 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
Hi eein,

I'm not really sure what you want to know here.  You state that the circle has a radius of 1, then there is a series of questions about estimating the value of 2d, d being the diameter.  Well, if the radius is 1, the diameters have to be 2, and 2d has to be 4, so I don't know what estimates you're referring to .

I think I've included below what I think you may be after here, in terms of the inscribed polygons and all:

For a 6-gon, each representative triangle has a 60 degree central angle, and is equilateral, so each side is 1, then the perimeter is 6.

For a 12-gon, each triangle has a central angle of 30.  If you draw an altitude to the outer side from the center, the smaller angle at the center is 15 degrees, and the side opposite it is given by x = sin 15 = 0.26.
This will be 1/2 the length of the side of the triangle, so each side will be 2 * 0.26 = 0.52, and the perimeter would be 6.211

If you keep doing this, you get the following:

   n(# sides)     P

       6          6

       12         6.211

       24         6.265


In general, in an n-gon, each side will be x = 2 sin(180/n)
and the perimeter will be n * 2 sin(180/n)

for a 2n-gon, each side will be y = 2 sin(180/2n) and the perimeter will be 2n * 2 sin(180/2n).

I'm not at all sure what you mean by "discuss the value of d"??

I hope at least some of this helped you.

Steve Holleran