In addition to toolic's and jethro's catches, note that the syntax of your formula for the area of a square isn't correct. It will give the area as (side times 2); you want $square_side**2 ($square_side to the power of 2, eg, squared).

Also, check your spelling (parallelogram and equilateral, for example), lest you teach li'l bro' errors and review your geometry text to check your title in the triangle formula.

Other suggestions, depending on your taste:

• When you wish to include info like your formulae in a non-executable format, include it inside POD markup, used as a block comment:
  

  =head formulae
  $area_square=$side**2;
  ...
  =cut
  [download]

• Unless you plan to expand this greatly, you'll have less overhead (and more portability) if you simply define pi as 3.141659 3.14159 (Thanks, eyepopslikeamosquito) rather than using Math::Trig.
• You may wish to provide the user with a choice to exit or restart. Various techniques are available.
• IMO using the equal sign in your answer lines (and generally, using the equals sign in any response to a user) risks making your code harder to proof ("Is that "=" an assignment or a symbol?"). I find usage such as print "Answer:  " . function(); or similar friendlier.

Updated: to acknowledge jethro's excellent point, posted as I fiddled with minutia and eyepopslikeamosquito's catch on my brain-fart.

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