Draw a unit circle.
Around it, circumscribe an equilateral triangle.
Circumscribe another circle around that.
Circumscribe a square around this circle.
Circumscribe another circle around that.
Circumscribe a regular pentagon around this circle.
Circumscribe another circle around that.

Continue, ad infinitum, with the next regular polygon.

Do the radii of these circles converge? If so, what is the limiting radius?

Each successive circle is sec(pi/n) times the radius of the one before it, or 1/cos(pi/n) times.

After the polygon with 10,000,000 sides, the radius of the outermost circle is 8.700032331912396 as found by

DEFDBL A-Z
Pi = 4 * ATN(1)
t = 1
FOR n = 3 TO 10000000
 t = t / COS(Pi / n)
 PRINT n, t
NEXT

And the last few lines (successive values) were:

9999983       8.700032331905089
9999984       8.700032331905518
9999985       8.700032331905948
9999986       8.700032331906378
9999987       8.700032331906808
9999988       8.700032331907238
9999989       8.700032331907668
9999990       8.700032331908098
9999991       8.700032331908528
9999992       8.700032331908957
9999993       8.700032331909387
9999994       8.700032331909817
9999995       8.700032331910247
9999996       8.700032331910677
9999997       8.700032331911107
9999998       8.700032331911537
9999999       8.700032331911967
10000000      8.700032331912396

so you can see where the increments are taking place from one iteration to the next.

I looked up 8.7000 and 8.70003 on
http://www.cecm.sfu.ca/projects/ISC/ISCmain.html

and found in the section

Bases; constants, Pi, e, sqrt(2), etc...
8700036625208194 = 1/Product(cos(Pi/n),n=3..infinity)

From the description, it sounds as if this is the most explicit way of expressing the result. (The site apparently leaves out the decimal point.)

 

Posted by Charlie on 2004-08-09 13:11:20