I am looking for a formula that can convert the volume of an octahedron to the length of an edge. So far, I have come across $\frac{1.442\cdot3\sqrt{v}}{1.122}$, but I am not sure if this equation is accurate. Thanks in advance!
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2$\begingroup$ Wikipedia to the rescue. $\endgroup$– vadim123Commented Jan 13, 2017 at 5:52
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$\begingroup$ I suspect that was supposed to be $\sqrt[3]v$, not $3\sqrt v$. But why $1.442/1.122$? Why not either give it as one decimal, or show its exact value (using a square root symbol)? You could edit the question to say where that value came from. (Do you have a link to a web page?) $\endgroup$– David KCommented Jan 13, 2017 at 7:57
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$\begingroup$ No, sorry. I forgot where i got it from. $\endgroup$– PMinerCommented Jan 20, 2017 at 22:45
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1 Answer
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See, that octahedron is dual to cube.
The volume of octahedron inscribed in cube is equal to $\frac{1}{6}$ volume of cube, ie: $$v_o = \frac{1}{6}v_c$$ And the length of a side of octahedron is equal to $\frac{\sqrt{2}}{2}$ side of cube, ie: $$l_o = \frac{\sqrt{2}}{2}l_c$$ Length of side of cube is equal to cubic root of it's volume: $$l_c = \sqrt[3]{v_c}$$
Then we have: $$l_o = \frac{\sqrt{2}}{2}\sqrt[3]{6v_o}$$