For future reference, you may want to give some thought to the way you name variables. For example, the variable name $radius_times_2 (Ln 4) seems to suggest that it should contain a value double that of some radius... but you then define it as $radius_times_2 = 3.141592654; or pi.
Your first var, $radius_times = 2; (Ln 3) is only a little bit better. The value 2 has no per se relevance to radius; it's merely a multiplier, so why not call it $multiplier?
By the time you get to $radius_times_2 *= $radius_times (Ln 5), you're waaaaay out in the deep grass. OTOH, something like $pi_times_2 = $pi * $multiplier is far less likely to trip the unwary than a redefinition of $pi. Yes, it's clumsy (there are many reformulations that would improve my offering) and it's less concise, but clarity trumps brevity in most practical cases.
You can't confuse the computer with that kind of thing, but you sure can make it hard on yourself; some future maintainer, or even the muzzy-brained, early-morning monk who considered offering assistance.
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