Diagonals of pentagon

Calculate the diagonal length of the regular pentagon:
a) inscribed in a circle of radius 12dm;
b) a circumscribed circle with a radius of 12dm.

Result

u1 =  22.825 dm
u2 =  28.214 dm

Solution:

$r_{ 1 }=12 \ \text{cm} \ \\ A=360/ (2 \cdot \ 5)=36 \ ^\circ \ \\ \ \\ \sin A=a_{ 1 }/2 : r_{ 1 } \ \\ \ \\ a_{ 1 }=2 \cdot \ r_{ 1 } \cdot \ \sin( A ^\circ \rightarrow\ \text{rad})=2 \cdot \ r_{ 1 } \cdot \ \sin( A ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ )=2 \cdot \ 12 \cdot \ \sin( 36 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ )=14.10685 \ \\ \ \\ u_{1}=u_{ 1 }=a_{ 1 }/2 \cdot \ (1+\sqrt{ 5 }) \doteq 14.1068/2 \cdot \ (1+\sqrt{ 5 }) \doteq 22.8254 \doteq 22.825 \ \text{dm}$
$r_{ 2 }=12 \ \text{cm} \ \\ \ \\ \tan A=a_{ 2 }/2 : r_{ 2 } \ \\ \ \\ a_{ 2 }=2 \cdot \ r_{ 2 } \cdot \ \tan( A ^\circ \rightarrow\ \text{rad})=2 \cdot \ r_{ 2 } \cdot \ \tan( A ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ )=2 \cdot \ 12 \cdot \ \tan( 36 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ )=17.43702 \ \\ \ \\ u_{2}=u_{ 2 }=a_{ 2 }/2 \cdot \ (1+\sqrt{ 5 }) \doteq 17.437/2 \cdot \ (1+\sqrt{ 5 }) \doteq 28.2137 \doteq 28.214 \ \text{dm}$

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