G'day WisDomSeeKer34,

Here's how I might tackle this:

#!/usr/bin/env perl

use strict;
use warnings;
use feature 'say';
use constant PI => 4 * atan2 1, 1;

say 'Type radius (>=0) or just hit Return to exit.';

while (1) {
    print 'Radius: ';
    chomp(my $radius = <>);
    last unless length $radius;

    if ($radius =~ /^[+-]?(?:\d+|\d*[.]\d+)$/) {
        if ($radius < 0) {
            say "Radius [$radius] is less than zero. Try again.";
            next;
        }

        say 'Circumference: ', 2 * PI * $radius;
    }
    else {
        say "Invalid radius [$radius]. Try again.";
        next;
    }
}
[download]

Notes:

• You don't need the utf8 pragma. Read its documentation to see what it's for; note the bold text telling you what it's not for.
• Consider whether you really want to load a module and all of its functions. As you only wanted pi, you could have loaded just that: "use Math::Trig 'pi';". It isn't really a huge deal with a tiny script like this; however, when you start writing more substantial code, it will matter. I didn't bother with a module for this at all: I just took the first example from the constant pragma.
• For this type of task, consider a loop that you can break out of (with last) or restart from somewhere in the middle (with next).
• Consider the user's exit condition. I've used "just hit Return"; typing "q", "quit", "exit" are all valid but makes the user do more work. In another scenario, hitting Return might take a default value, so something different is needed to exit.
• Note the use of print for the prompt (without newline); and the use of say for the feedback (with newline).
• You can add another 'last' after printing the circumference if you don't want this to keep looping. Seek advice from your users on this: if they're sitting there with a long list of radii, they may not appreciate having to type the command repeatedly for each one.
• The regex may look complicated but actually isn't. It's basically just an optional, leading sign followed by two alternatives (i.e. int or float). See perlre if this is unfamiliar.

Example output:

$ ./pm_1216483_loop_qanda.pl
Type radius (>=0) or just hit Return to exit.
Radius: 123.456
Circumference: 775.696925283163
Radius: q
Invalid radius [q]. Try again.
Radius: 0
Circumference: 0
Radius: +0
Circumference: 0
Radius: -0
Circumference: 0
Radius: 0.00000001
Circumference: 6.28318530717959e-08
Radius: +0.00000001
Circumference: 6.28318530717959e-08
Radius: -0.00000001
Radius [-0.00000001] is less than zero. Try again.
Radius: .00000001
Circumference: 6.28318530717959e-08
Radius: +.00000001
Circumference: 6.28318530717959e-08
Radius: -.00000001
Radius [-.00000001] is less than zero. Try again.
Radius: 1
Circumference: 6.28318530717959
Radius: -1
Radius [-1] is less than zero. Try again.
Radius: 1.
Invalid radius [1.]. Try again.
Radius: .1
Circumference: 0.628318530717959
Radius: load of rubbish
Invalid radius [load of rubbish]. Try again.
Radius: 
$
[download]