Triangular prism - regular

The regular triangular prism is 7 cm high. Its base is an equilateral triangle whose height is 3 cm. Calculate the surface and volume of this prism.

Correct answer:

V =  36.3731 cm³
S =  83.1384 cm²

Step-by-step explanation:

$$\begin{gathered} v=3\text{ cm } \\ h=7\text{ cm } \\ {a}^2=v^2+(a\mathrm{/}2{)}^2 \\ 4a^2=4v^2+a^2 \\ 3a^2=4v^2 \\ a=v\cdot 2\mathrm{/}\sqrt{3}=3\cdot 2\mathrm{/}\sqrt{3}=2\sqrt{3}\text{ cm}\doteq 3.4641\text{ cm } \\ {S}_1={\displaystyle \frac{\sqrt{3}}{4}}\cdot a^2={\displaystyle \frac{\sqrt{3}}{4}}\cdot 3.4641^2=3\sqrt{3}{\text{cm}}^2\doteq 5.1962{\text{cm}}^2 \\ V=S_1\cdot h=5.1962\cdot 7=21\sqrt{3}=36.3731{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} o=3\cdot a=3\cdot 3.4641=6\sqrt{3}\text{ cm}\doteq 10.3923\text{ cm } \\ S=2\cdot S_1+o\cdot h=2\cdot 5.1962+10.3923\cdot 7=48\sqrt{3}=83.1384{\text{cm}}^2 \end{gathered}$$

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