Triangular prism,

The regular triangular prism, whose edges are identical, has a surface of 2514 cm ^ 2 (square). Find the volume of this body in cm³ (l).

Correct result:

V =  7180.4472 cm³

Solution:

$$\begin{gathered} S=2514{\text{cm}}^2 \\ {S}_1=\sqrt{3}\mathrm{/}4a^2 \\ S=2\cdot S_1+3a\cdot a \\ S=\sqrt{3}\mathrm{/}2a^2+3a^2 \\ S=(\sqrt{3}\mathrm{/}2+3)a^2 \\ a=\sqrt{{\displaystyle \frac{S}{\sqrt{3}\mathrm{/}2+3}}}=\sqrt{{\displaystyle \frac{2514}{\sqrt{3}\mathrm{/}2+3}}}\doteq 25.5006\text{ cm } \\ {S}_1=\sqrt{3}\mathrm{/}4\cdot a^2=\sqrt{3}\mathrm{/}4\cdot 25.5006^2\doteq 281.5796 \\ V=S_1\cdot a=281.5796\cdot 25.5006\doteq 7180.4472=7180.4472{\text{cm}}^3 \\ \text{ Verifying Solution: } \\ {S}_2=2\cdot S_1+3\cdot a^2=2\cdot 281.5796+3\cdot 25.5006^2=2514{\text{cm}}^2 \\ {S}_2=S \end{gathered}$$

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