Dodecagon

A dodecagon is a 12-sided polygon. The sum of its internal angles is $1800^{\circ}$ .

A regular dodecagon can be seen below:

$[asy] for(int i = 0; i <= 11; ++i) { draw(dir(360/12*i)--dir(360/12*(i + 1))); } pair A,B,C,D,E,F,G,H,I,J,K,L; A=dir(360/12*0); B=dir(360/12*1); C=dir(360/12*2); D=dir(360/12*3); E=dir(360/12*4); F=dir(360/12*5); G=dir(360/12*6); H=dir(360/12*7); I=dir(360/12*8); J=dir(360/12*9); K=dir(360/12*10); L=dir(360/12*11); label("A",A,dir(0)); label("B",B,dir(30)); label("C",C,dir(60)); label("D",D,dir(90)); label("E",E,dir(120)); label("F",F,dir(150)); label("G",G,dir(180)); label("H",H,dir(210)); label("I",I,dir(240)); label("J",J,dir(270)); label("K",K,dir(300)); label("L",L,dir(330)); draw(dir(360/12*0)--dir(360/12*6)); dot((dir(360/12*0)+dir(360/12*6))/2); pair O = (dir(360/12*0)+dir(360/12*6))/2; label("O",O,S); draw(A--O); draw(Circle(O,1)); [/asy]$ The area of a regular dodecagon can be calculated by the formula $3R^2$ , where $R$ is the circumradius of the dodecagon. In this case, $R$ would be $OA$ . Also, each small triangle ( $AOB$ , $BOC$ , etc.) is congruent, so $\angle AOB=\angle BOC=\angle COD$ (etc) $=30^{\circ}$ .

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Dodecagon

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