A [regular](https://artofproblemsolving.com/wiki/index.php/Regular_polygon) 12-gon is inscribed in a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) of [radius](https://artofproblemsolving.com/wiki/index.php/Radius) 12.  The [sum](https://artofproblemsolving.com/wiki/index.php/Sum) of the lengths of all sides and [diagonals](https://artofproblemsolving.com/wiki/index.php/Diagonal) of the 12-gon can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers.  Find $a + b + c + d^{}_{}$.
[1990 AIME-12.png](https://artofproblemsolving.com/wiki/index.php/File:1990_AIME-12.png)
The easiest way to do this seems to be to find the length of each of the sides and diagonals. To do such, draw the [radii](https://artofproblemsolving.com/wiki/index.php/Radius) that meet the endpoints of the sides/diagonals; this will form [isosceles triangles](https://artofproblemsolving.com/wiki/index.php/Isosceles_triangle). Drawing the [altitude](https://artofproblemsolving.com/wiki/index.php/Altitude) of those [triangles](https://artofproblemsolving.com/wiki/index.php/Triangle) and then solving will yield the respective lengths.

The length of each of the 12 sides is $2 \cdot 12\sin 15$. $24\sin 15 = 24\sin (45 - 30) = 24\frac{\sqrt{6} - \sqrt{2}}{4} = 6(\sqrt{6} - \sqrt{2})$.
The length of each of the 12 diagonals that span across 2 edges is $2 \cdot 12\sin 30 = 12$ (or notice that the triangle formed is [equilateral](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle)).
The length of each of the 12 diagonals that span across 3 edges is $2 \cdot 12\sin 45 = 12\sqrt{2}$ (or notice that the triangle formed is a $45 - 45 - 90$ [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle)).
The length of each of the 12 diagonals that span across 4 edges is $2 \cdot 12\sin 60 = 12\sqrt{3}$.
The length of each of the 12 diagonals that span across 5 edges is $2 \cdot 12\sin 75 = 24\sin (45 + 30) = 24\frac{\sqrt{6}+\sqrt{2}}{4} = 6(\sqrt{6}+\sqrt{2})$.
The length of each of the 6 [diameters](https://artofproblemsolving.com/wiki/index.php/Diameter) is $2 \cdot 12 = 24$.
Adding all of these up, we get $12[6(\sqrt{6} - \sqrt{2}) + 12 + 12\sqrt{2} + 12\sqrt{3} + 6(\sqrt{6}+\sqrt{2})] + 6 \cdot 24$
$= 12(12 + 12\sqrt{2} + 12\sqrt{3} + 12\sqrt{6}) + 144 = 288 + 144\sqrt{2} + 144\sqrt{3} + 144\sqrt{6}$. Thus, the answer is $144 \cdot 5 = \boxed{720}$.