I found this is a puzzle book once and it intrigued me!

It starts off with a circle of radius 1m. That circle is enclosed with a regular triangle. This triangle is enclosed in another circle which is enclosed by a square... then a circle and then a pentagon, if you get the picture i.e. a circles are increasingly enclosed with regular polygons which increase their number of sides by 1.

The question is... as the number of sides of the regular polygons approach infinity, will the radius of the outer circle approach a definite limit or go on for ever.

I am unable to work out this sum on paper. I figured that the radius of each circle is equal to the radius of the previous circle divided by the sin of 1/2 the angle of the regular polygon.

As the angle of a regular polygon with sides n, is ((n-2)*180)/n)

Now let t = ((n-2)*180)/n)

Now using simple trigonometry, we see that Sin (t/2) = R(x)/R(x+1)

=> R(n+1) = R(n) / sin[t(1)/2]

=> R(n+2) = R(n+1) / sin[t(2)/2]

=> R(n+2) = [R(n) / sin[t(1)/2)] / sin[t(2)/2]

=> R(n+2) =R(n) / sin[t(1)/2)*sin(t(2)/2)]

I follows that as n approaches infinity the radius of the outermost circle will be the infinite product:-

R(inf) = R(1) / sin[t(1)/2)*sin(t(2)/2)*sin(t3)/2*............]

I hope this is clear, if you draw it out it will help...

Now my question is this:-, I did it by computer and it seems to approach 8.657m after the 1000 sided polygon is added, but it seems to increase by ever smaller amounts.Is there an explicit way to find this limit by hand

My second question is:-, I have lost the puzzle book I read it in but I remember reading the answer which was different from my result of about 8.657m.Have I made any mistakes with my reasoning