Triangular pyramid

What is the volume of a regular triangular pyramid with a side 3 cm long?

Correct result:

V =  3.182 cm³

Solution:

$$\begin{gathered} a=3\text{ cm } \\ {h}_1=\sqrt{a^2-(a\mathrm{/}2{)}^2}=\sqrt{3^2-(3\mathrm{/}2{)}^2}\doteq 2.5981\text{ cm } \\ h=\sqrt{h_1^2-(h_1\mathrm{/}3{)}^2}=\sqrt{2.5981^2-(2.5981\mathrm{/}3{)}^2}=\sqrt{6}\text{ cm}\doteq 2.4495\text{ cm } \\ {S}_1=a\cdot h_1\mathrm{/}2=3\cdot 2.5981\mathrm{/}2\doteq 3.8971{\text{cm}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 3.8971\cdot 2.4495=3.182{\text{cm}}^3 \end{gathered}$$

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