Dodecagon

Calculate the size of the smaller of the angles determined by lines A1 A4 and A2 A10 in the regular dodecagon A1A2A3. .. A12. Express the result in degrees.

Correct result:

x =  75 °

Solution:

A1=36012 1=30  A4=36012 4=120  A2=36012 2=60  A10=36012 10=300   s1=A4A12=120302=45  s2=A10A22=300602=120   x1=s2s1=12045=75  x2=180x1=18075=105   x=x1=75=75A_{1}=\dfrac{ 360 }{ 12 } \cdot \ 1=30 \ ^\circ \ \\ A_{4}=\dfrac{ 360 }{ 12 } \cdot \ 4=120 \ ^\circ \ \\ A_{2}=\dfrac{ 360 }{ 12 } \cdot \ 2=60 \ ^\circ \ \\ A_{10}=\dfrac{ 360 }{ 12 } \cdot \ 10=300 \ ^\circ \ \\ \ \\ s_{1}=\dfrac{ A_{4}-A_{1} }{ 2 }=\dfrac{ 120-30 }{ 2 }=45 \ ^\circ \ \\ s_{2}=\dfrac{ A_{10}-A_{2} }{ 2 }=\dfrac{ 300-60 }{ 2 }=120 \ ^\circ \ \\ \ \\ x_{1}=s_{2}-s_{1}=120-45=75 \ ^\circ \ \\ x_{2}=180 - x_{1}=180 - 75=105 \ ^\circ \ \\ \ \\ x=x_{1}=75=75 ^\circ



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