Find-the-volume-of-the-pyramid-which-is-folded-from-a-trangular-paper-with-sides-a-b-and-c-

Question Number 47195 by MrW3 last updated on 06/Nov/18

F i n d t h e v o l u m e o f t h e p y r a m i d w h i c h

i s f o l d e d f r o m a t r a n g u l a r p a p e r w i t h

s i d e s a, b a n d c .

Answered by MJS last updated on 06/Nov/18

in this case {it}^{'} s easy to use {Euler}^{'} s formula

because the sides of the bottom triangle

are \frac{a}{2}, \frac{b}{2}, \frac{c}{2} and the skew sides have the

same values

we get V = \frac{\sqrt{2}}{96} \sqrt{a^{4} (b^{2} + c^{2}) + b^{4} (a^{2} + c^{2}) + c^{4} (a^{2} + b^{2}) - (a^{6} + b^{6} + c^{6})} =

= \frac{\sqrt{2}}{96} \sqrt{(a^{2} + b^{2} - c^{2}) (a^{2} - b^{2} + c^{2}) (- a^{2} + b^{2} + c^{2})}

Commented by MrW3 last updated on 06/Nov/18

t h a n k s a l o t s i r!

Commented by ajfour last updated on 06/Nov/18

S i r p l e a s e e l a b o r a t e . .

Commented by MJS last updated on 07/Nov/18

Commented by MJS last updated on 07/Nov/18

a, b, c are the basic edges

p is skew to a

q is skew to b

r is skew to c

{Euler}^{'} s formula gives the volume of a general

tetrahedron

we know that V = \frac{B h}{3} with B being the area of

the basic triangle \Rightarrow h = \frac{3 V}{B} = \frac{ϵ}{4 B} with ϵ being

the root in {Euler}^{'} s formula

B = \frac{Δ}{4} with Δ = \sqrt{(a + b + c) (a + b - c) (a - b + c) (- a + b + c)}

\Rightarrow h = \frac{ϵ}{Δ}

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