polygons and circumscribing

[sombra]

I have two similar problems:
1) Draw a circle of radius r. Now circumscribe an equilateral triangle about the circle and then circumscribe another circle about the triangle and then circumscribe a square about that circle and continue indefinitely alternating circle and increasing the sides of a regular polygon. Find the relationship so that you can calculate the radius of the 5th circle, 10th circle, 20th circle etc.

2) Use regular polygons in the form of 2^2n for n = 1,2,... having a radius of r. Circumscribe the regular polygons running through n. There is a relationship between the area of the next polygon and the previous, find that formula.

I've worked quite a bit on these but I don't feel I am getting the answer that is expected. Help please?

 

[Deleted member 4993]

I have two similar problems:
1) Draw a circle of radius r. Now circumscribe an equilateral triangle about the circle and then circumscribe another circle about the triangle and then circumscribe a square about that circle and continue indefinitely alternating circle and increasing the sides of a regular polygon. Find the relationship so that you can calculate the radius of the 5th circle, 10th circle, 20th circle etc.

2) Use regular polygons in the form of 2^2n for n = 1,2,... having a radius of r. Circumscribe the regular polygons running through n. There is a relationship between the area of the next polygon and the previous, find that formula.

I've worked quite a bit on these but I don't feel I am getting the answer that is expected. Help please?

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

What relation ship did you find between the radii - when the circumscribing figures were triangle, square, pentagon and hexagon?

 

[sombra]

Hi I didn't find the relationship. I know if I use r as a unit then the radius of the first circle is 1 and then the second circle becomes 2 and the 3rd circle ? From the first to the 2nd circle I used the 30,60 90 triangle and the sin 30 gives me the length of the hypotenuse for the next radius being 2. I know the square has side length 4 so 1/2 the diagonal of the square is my next radius which is 2 sqrt2. I know my next polygon is a pentagon and I want the value of the segment from the center of the inner circle to any vertex. Part of my problem is that I can't visualize what happens with the pentagon with respect to the preceding circle and square. I know a radius from the center to a side of the pentagon will be one of my legs in a right triangle so I want the measure of the hypotenuse of this new right triangle which will be my new radius of the next circle. The angle across from the shorter side is 360/10 = 36 degrees. We know the length of the longer leg because it is the radius of the previous circle so the cos of 36 degrees is equal to the previous radius divided by the new radius. I know it is all here I just can't piece it together to write an expression relating it where I know I am correct. Help PLEASE? Is the relationship r_n = [r_(n-1)]/cos(180/2n)? I think so but I must have a recursion formula in order to find previous values. I mean for instance I can't find r_5 unless I know r_4. So let r_1 = 1 so r_2 = 1/cos 45 so r_3 = r_2/cos 36=1/[cos 45*cos36] or = 1/[cos(pi/4)*cos(pi/5)]. Ah I think this makes more sense because now I can relate the degrees to the number of sides as in 1/cos(pi/2n) for n>=2 with a product of the previous cos in the denominator. But I don't know how to express this?? How do I simplify writing a product of cosines??

New info. Ok here I go. Let n be the number of sides of the n-gon and define r_2 = 1 so r_3 = r_2/cos(pi/n) and r_n = r_2/[cos(pi/n)cos(pi/(n-1)...cos(pi/3)]. I still need help writing the product of the cosines compactly so I could find the value of r=100