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Question Number 134713 by bramlexs22 last updated on 06/Mar/21

$geometry$ All the edges of a regular square pyramid have a length of 8. What is the volume?\n
	Answered by john_santu last updated on 06/Mar/21

	$R e g u l e r s q u a r e p y r a m i d a l r e a d y$ $a r e g u l a r q u a d r i l a t e r a l .$ $A n e q u i l a t e r a l t r i a n g l e i s t w o o f$ $m i r r o r e d a c r o o s t h e l o n g l e g$ $30 ° - 60 ° - 90 ° r i g h t t h e t r i a n g l e s$ $w i t h s i d e s i n t h e r a t i o 1 : \sqrt{3} : 2,$ $T h e a l t i t u d e o f f a c e t r i a n g l e i s$ $\frac{\sqrt{3}}{2} \times 8 = 4 \sqrt{3} . T h a t t r i a n g l e i s$ $t i l t e d i n s o t h e t o p v e r t e x i s o v e r$ $t h e c e n t e r o f s q u a r e 4 u n i t s f r o m t h e$ $e d g e : h^{2} + 4^{2} = {(4 \sqrt{3})}^{2} \Rightarrow h = 4 \sqrt{2}$ $s o t h e V o l u m e = \frac{1}{3} A . h = \frac{1}{3} \times 8^{2} \times 4 \sqrt{2}$ $V o l = \frac{256 \sqrt{2}}{3} {u n i t s}^{3} ∙$
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