Each edge of a parallelepiped is 1 cm long. At one of its vertices, all three face angles are acute, and each measures 2α. Find the volume of the parallepiped. Help me, please


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Question Number 218162 by Ismoiljon_008 last updated on 31/Mar/25

$E a c h e d g e o f a p a r a l l e l e p i p e d i s 1 c m l o n g .$ $A t o n e o f i t s v e r t i c e s, a l l t h r e e f a c e a n g l e s$ $a r e a c u t e, a n d e a c h m e a s u r e s 2 α .$ $F i n d t h e v o l u m e o f t h e p a r a l l e p i p e d .$ $H e l p m e, p l e a s e$
	Answered by mr W last updated on 31/Mar/25

	Commented by mr W last updated on 31/Mar/25

	$a = b = c = 1$ $b a s e a r e a A_{B a s e} = a b \sin 2 α = \sin 2 α$ $h = \sqrt{1^{2} - {(\frac{1 \times \cos 2 α}{\cos α})}^{2}} = \sqrt{1 - \frac{\cos^{2} 2 α}{\cos^{2} α}}$ $V = A_{B a s e} h = \sin 2 α \sqrt{1 - \frac{\cos^{2} 2 α}{\cos^{2} α}} = 2 \sin^{2} α$
	Commented by Ismoiljon_008 last updated on 31/Mar/25

	$T h a n k y o u v e r y m u c h$
	Commented by Ismoiljon_008 last updated on 31/Mar/25

	$M r W, w h y i s t h e l e n g t h o f A D e q u a l$ $t o \frac{1 \cdot c o s 2 α}{c o s α} ?$
	Commented by mr W last updated on 31/Mar/25

	$A C = A B \cos 2 α = 1 \times \cos 2 α$ $A D = \frac{A C}{\cos α} = \frac{1 \times \cos 2 α}{\cos α}$
	Commented by Ismoiljon_008 last updated on 31/Mar/25

	$I g o t i t . T h a n k y o u M r W$
	Commented by mr W last updated on 31/Mar/25

	$t h e g e n e r a l f o r m u l a :$
	Commented by mr W last updated on 31/Mar/25

	Commented by Ismoiljon_008 last updated on 01/Apr/25

	$T h a n k s$
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