1.Given a regular tetrahedron ABCD with vertices A(0,0,0)B(a,0,0), C(0,a,0),and D(0,0,a).Calculate the volume V and the surface area S of this tetrahedron.


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Question Number 211603 by MrGaster last updated on 14/Sep/24

$1 . Given a regular tetrahedron A B C D$ $with vertices A (0, 0, 0) B (a, 0, 0),$ $C (0, a, 0), and D (0, 0, a) . Calculate the$ $volume V and the surface area S of$ $this tetrahedron .$
	Answered by BHOOPENDRA last updated on 14/Sep/24

	Answered by BHOOPENDRA last updated on 15/Sep/24

	$T h e v o l u m e o f a t e t r a h e d r o n w i t h$ $v e r t i c e s A, B, C, & D i s g i v e n f o r m u l a$ $V = \frac{1}{6} ∣ A . (B \times C) ∣$ $A (0, 0, 0), B (a, 0, 0), C (0, a, 0) D (0, 0, a)$ $V = \frac{1}{6} \| \begin{matrix} 0 0 0 \\ a 0 0 \\ 0 a 0 \\ 0 0 a \end{matrix} \| = \frac{1}{6} ∣ a^{3} ∣$ $V = \frac{1}{6} a^{3}$ $S u r f a c e a r e a = (3 \times a r e a r i g h t a n g l e t r i a n g l e) + e q u i l e t r a l t r i a n g l e a r e a$ $= 3 \times \frac{1}{2} a^{2} + \frac{\sqrt{3}}{4} {(\sqrt{2} a)}^{2}$ $= \frac{(3 + \sqrt{3}) a^{2}}{2}$
	Commented by BHOOPENDRA last updated on 14/Sep/24

	$I t c a n b e a l s o s o l v e d b y u s i n g$ $i n t e g r a t i o n$
	Answered by mr W last updated on 15/Sep/24

	$V = \frac{b a s e a r e a \times h e i g h t}{3} = \frac{0.5 a^{2} \times a}{3} = \frac{a^{3}}{6}$ $S = 3 \times \frac{a^{2}}{2} + \frac{\sqrt{3} {(\sqrt{2} a)}^{2}}{4} = \frac{(3 + \sqrt{3}) a^{2}}{2}$
	Commented by BHOOPENDRA last updated on 15/Sep/24

	$T h a n k s M r . W i h a v e c o r r e c t e d$
	Commented by mr W last updated on 15/Sep/24
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