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Zadatak 0118

Test: MPsIIIJAI_023

Izračunati zapreminu pravilnog tetraedra u funkciji osnovne ivice a.

Rešenje:

\(V=\frac{1}{3}\cdot \frac{a^2\sqrt{3}}{4}\cdot H\)

\(H^2=\left ( \frac{a\sqrt{3}}{2} \right )^2-\left ( \frac{a\sqrt{3}}{6} \right )^2\)

\(H^2= \frac{3a^2}{4} -\frac{3a^2}{6}\)

\(H^2= \frac{24a^2}{36} =\frac{2a^2}{3}\)

\(H=a\sqrt{\frac{2}{3}}\);

\(V=\frac{1}{12}\cdot a^2\sqrt{3}\cdot a\frac{\sqrt{2}}{\sqrt{3}}\)

\(V=\frac{a^2\sqrt{2}}{12}\).

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