Question

Calculați volumul unei piramide patrulatere regulate cu toate muchiile de lungime 2a.

Answer 1

Fie piramida ABCD si O centrul bazei.
$AO= \frac{AC}{2}= \frac{AB \sqrt{2} }{2}= \frac{2a \sqrt{2} }{2}= a\sqrt{2}.$
Aplic teorema lui pitagora in ΔVOA:
$AO^{2}+ OV^{2} =AV^{2}=>OV^{2} = AV^{2}-AO^{2} => OV= \sqrt{ AV^{2}- AO^{2} } \\ OV= \sqrt{4a^{2}-2 a^{2} }= \sqrt{2a^{2}} =a \sqrt{2}$
$A_{ABCD}= AB^{2}= 4a^{2}$
$V_{VABCD} =VO* A_{ABCD}= a \sqrt{2}* 4a^{2}= 4a^{3} \sqrt{2}.$