Five-gon

Calculate the side a, the circumference and the area of the regular 5-angle if Rop = 6cm.

Correct result:

a = 7.0534 cm
o = 35.2671 cm
S = 85.5951 cm²

Solution:

n = 5 R = 6 cm Φ = 360 / (2 \cdot n) = 360 / (2 \cdot 5) = 3 6^{\circ} \sin Φ = a / 2 : R a = 2 \cdot R \cdot \sin Φ^{\circ} = 2 \cdot R \cdot \sin 3 6^{\circ} = 2 \cdot 6 \cdot \sin 3 6^{\circ} = 2 \cdot 6 \cdot 0.587785 = 7.053 = 7.0534 cm

o = n \cdot a = 5 \cdot 7.0534 = 35.2671 cm

h = \sqrt{R^{2} - (a / 2)^{2}} = \sqrt{6^{2} - (7.0534 / 2)^{2}} ≐ 4.8541 cm S_{1} = \frac{a \cdot h}{2} = \frac{7.0534 \cdot 4.8541}{2} ≐ 17.119 {cm}^{2} S = n \cdot S_{1} = 5 \cdot 17.119 = 85.5951 {cm}^{2}

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