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Let $s$ be the edge length of a regular dodecahedron. As a function of $s$, what is the dodecahedron's minimum and maximum diagonal (i.e. cross-section)?

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    @Joseph Malkevitch, yes, I do allow diagonals within a face - this should allow the shortest possible diagonal across the dodecahedron.2011-07-17

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For unit edge length, the longest diagonal is 2.802517076888147. If this is not what you mean (as joriki's comment question indicates), my apologies.
Dodecahedron

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    @Joseph, for fun, plugging in your value of "2.802517076888147" into the inverse symbolic calculator generates:"(5+5^(1/2))*15^(1/2)" which equal to exactly 1/10th the exact expression for the longest diagonal. =)2011-07-17
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In this post of mine I posted this link to a small notebook (see section 15.3. Dodecaedro, in Portuguese), where I compute the relation

$\frac{l}{d}=\frac{1}{2\sqrt{3}\cos \left( \frac{\pi }{5}\right) }=\frac{2}{% \sqrt{3}\left( 1+\sqrt{5}\right) }\approx 0.356\,82,$

where $l$ is the edge length (denoted $s$ by you) and $d$ is the diagonal length connecting two opposite vertices.

So

$\frac{d}{l}=2\sqrt{3}\cos \left( \frac{\pi }{5}\right) =\frac{\sqrt{3}% \left( 1+\sqrt{5}\right) }{2}\approx 2.802\,5.$

enter image description here

Top figure: edges (black), diagonal (red), axes of faces (magenta). Bottom figure: cross-section rotated with respect to the top figure.

It you want the cross-section area, then I think the bottom figure may help. One would have to compute the height (brown), the radius of the circumscribed circle to the faces, and its apothem.

As far as the other "diagonals" I have not computed them.

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    @Roger Harris: You are welcome!2011-07-17
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Wikipedia gives the radius of the circumscribed sphere as $\frac{\sqrt{15}+\sqrt{3}}{4}s\approx 1.401s$. Your maximum diameter is twice this. It also gives the radius of the inscribed sphere as $\frac{\sqrt{250+110\sqrt{5}}}{20}s\approx 1.1135s$. Is this whatyou menant by minimum diagonal-it is between two face centers.

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    @Ross, perfect, this is exactly what I was looking for. I can't believe I missed the wikipedia article...2011-07-17