Cyclic polygons. (See if you can do this without a calculating aid)

$A_{ n }=\frac { n }{ 2 } \sin (\frac { 2\pi }{ n } ).\\ As\quad n\quad tends\quad to\quad infinity,\quad A_{ n }\quad tends\quad to\quad 3.141...\\ So,\quad \left\lfloor A_{ n } \right\rfloor =3.\quad But\quad when\quad does\quad this\quad start\quad happening?\\ Set\quad A_{ n }=3\quad and\quad we\quad get\quad n\ge 6.\quad Also\quad notice\quad that\quad the\quad RHS\\ is\quad rational.\quad So,\quad we\quad need\quad to\quad get\quad the\quad minimum\quad value\quad of\\ n\quad such\quad that\quad 6/n\quad is\quad rational\quad and\quad less\quad than\quad 1.\quad So,\quad we\quad get\\ n=12.\quad For\quad n\ge 13,\quad we\quad have\quad \left\lceil A_{ n } \right\rceil \quad =\quad 4\quad =\quad \left\lceil A_{ 99 } \right\rceil .\quad \\ So,\quad b=13\quad and\quad a=3.\quad a+b=\boxed { 16. }$