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.

Final Answer:

u₁ =  22.8254 dm
u₂ =  28.2137 dm

Step-by-step explanation:

$$\begin{gathered} r_1=12\text{cm} \\ A=360/(2\cdot 5)=36{}^{\circ } \\ \sin A=a_1/2:r_1 \\ {a}_1=2\cdot r_1\cdot \sin A=2\cdot r_1\cdot \sin 36°=2\cdot 12\cdot \sin 36°=2\cdot 12\cdot 0.587785=14.10685\text{cm} \\ {u}_1=a_1/2\cdot (1+\sqrt{5})=14.1068/2\cdot (1+\sqrt{5})=22.8254\text{dm} \end{gathered}$$

$$\begin{gathered} r_2=12\text{cm} \\ \tan A=a_2/2:r_2 \\ {a}_2=2\cdot r_2\cdot \tan A=2\cdot r_2\cdot \tan 36°=2\cdot 12\cdot \tan 36°=2\cdot 12\cdot 0.726543=17.43702\text{cm} \\ {u}_2=a_2/2\cdot (1+\sqrt{5})=17.437/2\cdot (1+\sqrt{5})=28.2137\text{dm} \end{gathered}$$

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