3. At n = 400, there are three decimal places of accuracy in both the Inscribed and Circumscribed cases. The average of the two values yields four decimal places of accuracy.

The following Mathematica code will generate a table of values of approximations of Pi for n = 4 to 400 in steps of 22. The table offers the Inner and Outer polygon estimates. The Average Error in the table is
Abs[(Inner + Outer)/2]

Input := 

TableForm[ 
Table[ {n,N[n/2 Sin[2 Pi/n],10],N[n Tan[Pi/n],10],
 N[Abs[(n/2 Sin[2 Pi/n] + n Tan[Pi/n])/2], 10]},
		{n,4,400,22}],
		TableHeadings->{{},{"n", "Inner(n)", "Outer(n)",
		  "Average Area"}}]

Output =

   n     Inner(n)      Outer(n)      Average Area
   4     2.            4.            3.

   26    3.111103636   3.156971564   3.1340376

   48    3.132628613   3.146086215   3.139357414

   70    3.137375812   3.143703625   3.140539718

   92    3.139151015   3.142814328   3.140982671

   114   3.14000234    3.142388173   3.141195256

   136   3.140475188   3.142151565   3.141313377

   158   3.140764693   3.142006732   3.141385712

   180   3.140954703   3.141911687   3.141433195

   202   3.141086089   3.141845973   3.141466031

   224   3.141180703   3.141798653   3.141489678

   246   3.141251088   3.141763453   3.141507271

   268   3.141304863   3.141736561   3.141520712

   290   3.14134687    3.141715554   3.141531212

   312   3.141380309   3.141698832   3.141539571

   334   3.141407361   3.141685305   3.141546333

   356   3.141429554   3.141674207   3.141551881

   378   3.141447987   3.14166499    3.141556488

   400   3.141463462   3.141657252   3.141560357

Input := 

N[Pi,10]

Output =

3.141592654

Input := 

TableForm[ 
Table[ {n,
 N[Abs[Pi - n/2 Sin[2 Pi/n]]], N[Abs[Pi - n Tan[Pi/n]]],
		N[Abs[(n/2 Sin[2 Pi/n] + n Tan[Pi/n])/2], 10]},
		{n,4,400,22}],
		TableHeadings->{{},{"n", "Inner Error",
		"Outer Error", "Average Error"}}]

Output =

   n     Inner Error   Outer Error    Average Error
   4     1.14159       0.858407       3.

   26    0.030489      0.0153789      3.1340376

   48    0.00896404    0.00449356     3.139357414

   70    0.00421684    0.00211097     3.140539718

   92    0.00244164    0.00122167     3.140982671

   114   0.00159031    0.000795519    3.141195256

   136   0.00111747    0.000558912    3.141313377

   158   0.000827961   0.000414079    3.141385712

   180   0.00063795    0.000319033    3.141433195

   202   0.000506564   0.000253319    3.141466031

   224   0.000411951   0.000206       3.141489678

   246   0.000341565   0.000170799    3.141507271

   268   0.000287791   0.000143907    3.141520712

   290   0.000245783   0.0001229      3.141531212

   312   0.000212344   0.000106179    3.141539571

   334   0.000185292   0.0000926511   3.141546333

   356   0.000163099   0.0000815534   3.141551881

   378   0.000144667   0.0000723364   3.141556488

   400   0.000129191   0.000064598    3.141560357