There are so much good points shown by other monks, but I have an other tip for you:

instead of

my $pi=3.141592654;

you could use

use constant PI => 3.141592654;

(in case you don't want to change PI in your program ;) )

Update: And you could use

print << 'END', "-" x76, "\n";
Welcome to Will's Plane Geometry Calculator.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The first argument should be tri, cir, rec, or sqr.

The second argument should be area, peri, or hypo.

If you give hypo as the second argument, the next two
arguments should be the known sides of the traingle.

The peri arg will also find circumference if cir is
given for the figure. Specify the diameter to find this.

Pi is approximated at 3.141592654 (simply change my pi
if a different approximation is needed).

To find area of a circle specify the radius.

I think you can figure out the rest.
END
[download]

That is much clearer to read for me.

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