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The method I use below (integrate its surface area) works for every regular polytope, so it could be the method Wolfram Alpha uses. The only difficult part is simplification of a complicated trigonometric expression, but Wolfram Alpha eats those for breakfast.
On the other hand there are only three families of regular polytope above dimension 4, so maybe it just knows a general formula for those three families and then just has the five exceptional regular polytopes programmed in as special cases.
The second of those seems extremely likely. (But, I repeat, I don't really know anything about this stuff.)