Question 1087817
<pre>
Here's how a regular dodecahedron looks:
 *[illustration Dodecahedron.jpg]
All 12 faces of a regular dodecahedron are 
congruent regular pentagons:  

{{{drawing(200,200,-1.5,1.5,-1.5,1.5,
line(0.95105652,0.30901699,0,1),
line(0,1,-0.95105652,0.30901699),
line(-0.95105652,0.30901699,-0.58778525,-0.80901699),
line(-0.58778525,-0.80901699,0.58778525,-0.80901699),
line(0.58778525,-0.80901699,0.95105652,0.30901699),
locate(-.2,-0.80901699,edge)

 )}}}

The area of a regular pentagon is given by the formula

{{{A=(
(
sqrt(5(5+2sqrt(5)))
)
/4

)*(edge)}}}

[If you don't know this formula, then ask me how to get it
in the thank-you note form below and I'll get back to you
by email.]

All 12 faces of the regular dodecahedron together have area 12
times that or

{{{3sqrt(5(5+2sqrt(5)))*edge}}}

Thus in this case,

{{{3sqrt(5(5+2sqrt(5)))*edge}}}{{{""=""}}}{{{288}}}

{{{sqrt(5(5+2sqrt(5)))*edge}}}{{{""=""}}}{{{96}}} 

{{{edge}}}{{{""=""}}}{{{96/sqrt(5(5+2sqrt(5)))}}}

Rationalizing the denominator gives:

{{{edge}}}{{{""=""}}}{{{expr(96/5)sqrt(5-2sqrt(5))}}}

Edwin</pre>