# Hexagonal prism 2

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

Correct result:

V =  20.175 cm3

#### Solution:

$h=5 \ \\ S=140 \ \\ S=2 \cdot \ 6 \cdot \ S_{1} + 6ah \ \\ S=2 \cdot \ 6 \cdot \ a^2 \cdot \ \sqrt{ 3 }/4 + 6ah \ \\ 140=3 \cdot \ \sqrt{ 3 } \cdot \ a^2 + 30a \ \\ -5.19615242271a_{2}-30a+140=0 \ \\ D=b_{2} - 4ac=3809.84535672 \ \\ \ \\ a_{1}=-8.8261406997 \doteq -8.8261 \ \\ a_{2}=3.05263800781 \doteq 3.0526 \ \\ a=a_{2}=3.0526 \doteq 3.0526 \ \\ S_{1}=a^2 \cdot \ \sqrt{ 3 }/4=3.0526^2 \cdot \ \sqrt{ 3 }/4 \doteq 4.0351 \ \\ S_{2}=6 \cdot \ a \cdot \ h=6 \cdot \ 3.0526 \cdot \ 5 \doteq 91.5791 \ \\ S_{3}=2 \cdot \ 6 \cdot \ S_{1}+S_{2}=2 \cdot \ 6 \cdot \ 4.0351+91.5791=140 \ \\ V=S_{1} \cdot \ h=4.0351 \cdot \ 5=20.175 \ \text{cm}^3$

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