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-

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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