Hexagonal prism 2

The regular hexagonal prism has a surface of 140 cm² and a height of 5 cm. Calculate its volume.

Correct answer:

V = 121.0521 cm³

h = 5 cm S = 140 cm^{2} S = 2 \cdot 6 \cdot S_{1} + 6 a h S = 2 \cdot 6 \cdot a^{2} \cdot s q r t (3) / 4 + 6 \cdot a \cdot h 140 = 2 \cdot 6 \cdot a^{2} \cdot 3 / 4 + 6 \cdot a \cdot 5 - 5.1961524227066 a^{2} - 30 a + 140 = 0 5.1961524227066 a^{2} + 30 a - 140 = 0 p = 5.196152; q = 30; r = - 140 D = q^{2} - 4 p r = 3 0^{2} - 4 \cdot 5.196152 \cdot (- 140) = 3809.8453567157 D > 0 a_{1, 2} = \frac{- q \pm D}{2 p} = \frac{- 3 0 \pm 3 8 0 9 . 8 5}{1 0 . 3 9 2 3 0 5} a_{1, 2} = - 2.886751 \pm 5.939389 a_{1} = 3.052638008 a_{2} = - 8.8261407 a = a_{1} = 3.0526 ≐ 3.0526 cm S_{1} = a^{2} \cdot 3 / 4 = 3.052 6^{2} \cdot 3 / 4 ≐ 4.0351 cm^{2} S_{2} = 6 \cdot a \cdot h = 6 \cdot 3.0526 \cdot 5 ≐ 91.5791 cm^{2} P = 6 \cdot S_{1} = 6 \cdot 4.0351 ≐ 24.2104 cm^{2} V = P \cdot h = 24.2104 \cdot 5 ≐ 121.0521 cm^{3} Verifying Solution: S_{3} = 2 \cdot P + S_{2} = 2 \cdot 24.2104 + 91.5791 = 140 cm^{2}

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