Zadatak 0118
- Detalji
- Kategorija: Površine i zapremine rogljastih tela
- Objavljeno 10 avgust 2013
- Autor Super User
- Pogodaka: 1006
Test: MPsIIIJAI_023
Izračunati zapreminu pravilnog tetraedra u funkciji osnovne ivice a.
Rešenje:
\(V=\frac{1}{3}\cdot \frac{a^2\sqrt{3}}{4}\cdot H\)
\(H^2=\left ( \frac{a\sqrt{3}}{2} \right )^2-\left ( \frac{a\sqrt{3}}{6} \right )^2\)
\(H^2= \frac{3a^2}{4} -\frac{3a^2}{6}\)
\(H^2= \frac{24a^2}{36} =\frac{2a^2}{3}\)
\(H=a\sqrt{\frac{2}{3}}\);
\(V=\frac{1}{12}\cdot a^2\sqrt{3}\cdot a\frac{\sqrt{2}}{\sqrt{3}}\)
\(V=\frac{a^2\sqrt{2}}{12}\).