4. The easiest method of solution is by trial and error (see note in ISSUES IN SOLUTIONS.)

4 digits - Inscribed case: 473

Input := 

N[n/2 Sin[2 Pi/n] /. n -> 473, 10]
N[n/2 Sin[2 Pi/n] /. n -> 472, 10]
N[Pi, 10]

Output =

3.141500262

Output =

3.14149987

Output =

3.141592654

So, n = 473 is the smallest such n.

4 digits - Circumscribed case: 1187

Input := 

N[n Tan[Pi/n] /. n -> 1186, 10]
N[n Tan[Pi/n] /. n -> 1187, 10]
N[Pi, 10]

Output =

3.141600001

Output =

3.141599989

Output =

3.141592654

So, n = 1187 is the smallest such n.

6 digits - Inscribed case: 5624

Input := 

N[n/2 Sin[2 Pi/n] /. n -> 5623, 15]
N[n/2 Sin[2 Pi/n] /. n -> 5624, 15]
N[Pi, 10]

Output =

3.14159199982404

Output =

3.14159200005651

Output =

3.141592654

So, n = 5624 is the smallest such n.

6 digits - Circumscribed case: 5463

Input := 

N[n Tan[Pi/n] /. n -> 5462, 15]
N[n Tan[Pi/n] /. n -> 5463, 15]
N[Pi, 10]

Output =

3.1415930000274

Output =

3.14159299990058

Output =

3.141592654

So, n = 5463 is the smallest such n.