A few suggestions:

• Break the Law_Of_Sines() function up a bit more. For example, you have a (rather hefty) section of code cut'n'pasted three times. Make it a subroutine.
• Instead of running $A, $a, $B, $b, $C, and $c, why not do a list of hashes called angles? So, you could do something like $angles->[0]{B} to reference the second angle in the first triangle. The real benefit of that is to allow you to do for loops, reducing the number of lines while improving readability.

An example of the second suggestion would be:

    unless ( $A > 0 && $B > 0 && $C > 0 ) {
        if ( $A > 0 && $B > 0 ) { $var = "C" }
        if ( $A > 0 && $C > 0 ) { $var = "B" }
        if ( $B > 0 && $C > 0 ) { $var = "A" }
        if ( $var eq "C" ) {
            $sum = $angles{B} + $angles{A};
            $other = 180 - $sum;
            $angles{$var} = $other;
            $C = sprintf( "%.10f", Math::NumberCruncher::deg2rad( $oth
+er ) );
        } elsif ( $var eq "B" ) {
            $sum = $angles{A} + $angles{C};
            $other = 180 - $sum;
            $angles{$var} = $other;
            $B = sprintf( "%.10f", Math::NumberCruncher::deg2rad( $oth
+er ) );
        } elsif ( $var eq "A" ) {
            $sum = $angles{B} + $angles{C};
            print ("Sum: $sum\n");
            $other = 180 - $sum;
            $angles{$var} = $other;
            $A = sprintf( "%.10f", Math::NumberCruncher::deg2rad( $oth
+er ) );
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instead could be written as:

if (grep $_ <= 0, @{$angles->[0]}{('A' .. 'C')}) {
    my $zero_angle = 'A';
       if ($angles->[0]{C} <= 0) { $zero_angle = 'C' }
    elsif ($angles->[0]{B} <= 0) { $zero_angle = 'B' }

    my @sides = ('A' .. 'C');
    @sides = grep $_ ne $zero_angle, @sides;

    $sum = 0;
    $sum += $_ for @{$angles->[0]}{@sides};
    $other = 180 - $sum;
    $angles{$var} = $other;
    $angles->[0]{$zero_angle} = 
        sprintf( "%.10f", Math::NumberCruncher::deg2rad( $other ) );
}
[download]

The primary difference between your version and mine is that I chose a data structure that allows me to have the interpreter associate pieces of data that should be associated together. Your version requires that you continually associate them yourself.

The second difference is that I'm taking advantage of the easy conversions between hashes and arrays through hash-slices. Read up on them. They're SOOOO powerful. :)

Now, you don't have to use my data structure. I chose it because it was the closest to what you were doing. However, it can easily make sense to have the angles in an array and not a hash. So, triangle 1, angle 2 would be $angles->[0][1]. Simple enough.

Keep with it. It's a neat idea!

Update: Fixed a few square brackets. Thanx Albannach and ChemBoy!