Difference between revisions of "Dodecagon"
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| − | A '''dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\circ}</math>. | + | A '''dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\circ}</math>. Each of its exterior angles has measure <math>30^{\circ}</math>. |
A regular dodecagon can be seen below: | A regular dodecagon can be seen below: | ||
| − | + | <asy> | |
for(int i = 0; i <= 11; ++i) { | for(int i = 0; i <= 11; ++i) { | ||
draw(dir(360/12*i)--dir(360/12*(i + 1))); | draw(dir(360/12*i)--dir(360/12*(i + 1))); | ||
| Line 36: | Line 36: | ||
pair O = (dir(360/12*0)+dir(360/12*6))/2; | pair O = (dir(360/12*0)+dir(360/12*6))/2; | ||
label("O",O,S); | label("O",O,S); | ||
| − | draw(A-- | + | draw(A--O); |
draw(Circle(O,1)); | draw(Circle(O,1)); | ||
| − | + | </asy> | |
| − | The area of a regular dodecagon can be calculated by the formula <math>3R^2</math>, where <math>R</math> is the circumradius of the dodecagon. In this case, <math>R</math> would be <math>OA</math>. | + | The area of a regular dodecagon can be calculated by the formula <math>3R^2</math>, where <math>R</math> is the circumradius of the dodecagon. In this case, <math>R</math> would be <math>OA</math>. Also, each small triangle (<math>AOB</math>, <math>BOC</math>, etc.) is congruent, so <math>\angle AOB=\angle BOC=\angle COD</math> (etc) <math>=30^{\circ}</math>. |
==See Also== | ==See Also== | ||
Latest revision as of 12:08, 15 June 2018
A dodecagon is a 12-sided polygon. The sum of its internal angles is 
. Each of its exterior angles has measure 
.
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]](http://latex.artofproblemsolving.com/0/e/1/0e12fea08044e8c9b01f003b869951661635201d.png)
The area of a regular dodecagon can be calculated by the formula 
, where 
is the circumradius of the dodecagon. In this case, would be 
. Also, each small triangle (
, 
, etc.) is congruent, so 
(etc) 
.
See Also
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