ISSUES IN SOLUTION

By asking for decimal places of accuracy, we make the problem easier to do by trial and error. Note that, say, 5 decimal places of accuracy does not mean that the error is < 0.000009. See the example below.

Input := 

FindRoot[ Pi - n/2 Sin[2 Pi/n] == 0.000009, {n, 1000}]

Output =

{n -> 1512.95}

Input := 

n5 = Floor[n /. %] + 1

Output =

1513

Input := 

N[n/2 Sin[2 Pi/n] /. n -> n5, 10]
N[Pi, 10]

Output =

3.141583624

Output =

3.141592654

So, n5 = 1513 doesn't even give 5 decimal places of accuracy, but it gives a starting place.

Input := 

N[n/2 Sin[2 Pi/n] /. n -> 2790, 10]

Output =

3.141589998

Input := 

N[n/2 Sin[2 Pi/n] /. n -> 2791, 10]

Output =

3.14159

So, n5 = 2791 is the smallest integer yielding 5 decimal places of accuracy; the difference is given below.

Input := 

N[Pi - n/2 Sin[2 Pi/n] /. n -> 2791, 10]

Output =

              -6
2.653619049 10

It's interesting to note that the circumscribed case starts off requiring larger n-values to well approximate Pi to a fixed number of digits, but it rapidly over takes the inscribed case. Of course, if only considering absolute error, the circumscribed case is always better.
^*)