Hexagon rotation

A regular hexagon of side 6 cm is rotated at 60° along a line passing through its longest diagonal. What is the volume of the figure thus generated?

Correct answer:

V =  226.1947 cm³

Step-by-step explanation:

$$\begin{gathered} a=6\text{cm} \\ k=60/180=\frac{1}{3}\doteq 0.3333 \\ h=\sqrt{a^2-(a/2{)}^2}=\sqrt{6^2-(6/2{)}^2}=3\sqrt{3}\text{cm}\doteq 5.1962\text{cm} \\ {S}_1=\pi \cdot h^2=3.1416\cdot 5.1962^2\doteq 84.823{\text{cm}}^2 \\ {V}_1=S_1\cdot a=84.823\cdot 6\doteq 508.938{\text{cm}}^3 \\ {V}_2=1/3\cdot S_1\cdot (a/2)=1/3\cdot 84.823\cdot (6/2)\doteq 84.823{\text{cm}}^3 \\ V=k\cdot (V_2+V_1+V_2)=0.3333\cdot (84.823+508.938+84.823)=226.1947{\text{cm}}^3 \end{gathered}$$

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