Regular hexagonal prism

Calculate the volume of a regular hexagonal prism whose body diagonals are 24cm and 25cm long.

Correct result:

V =  2636.7965 cm³

Solution:

$$\begin{gathered} u=24\text{ cm } \\ v=25\text{ cm } \\ {u}_1=2a \\ (u_2\mathrm{/}2{)}^2+(a\mathrm{/}2{)}^2=a^2 \\ {u}_2^2\mathrm{/}4+a^2\mathrm{/}4=a^2 \\ {u}_2^2\mathrm{/}4+=3\mathrm{/}4a^2 \\ {u}_2=\sqrt{3}\cdot a \\ {h}^2+u_1^2=v^2 \\ {h}^2+u_2^2=u^2 \\ {h}^2+(2a{)}^2=v^2 \\ {h}^2+(\sqrt{3}\cdot a{)}^2=u^2 \\ 4a^2-3a^2=v^2-u^2 \\ {a}^2=v^2-u^2 \\ a=\sqrt{v^2-u^2}=\sqrt{25^2-24^2}=7\text{ cm } \\ h=\sqrt{v^2-4\cdot a^2}=\sqrt{25^2-4\cdot 7^2}=\sqrt{429}\text{ cm}\doteq 20.7123\text{ cm } \\ {S}_1={\displaystyle \frac{\sqrt{3}\cdot a^2}{4}}={\displaystyle \frac{\sqrt{3}\cdot 7^2}{4}}\doteq 21.2176{\text{cm}}^2 \\ S=6\cdot S_1=6\cdot 21.2176\doteq 127.3057 \\ V=S\cdot h=127.3057\cdot 20.7123=2636.7965{\text{cm}}^3 \end{gathered}$$

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