How does the area of a regular n-gon compare to the area of a non-regular n-gon? For triangles, the formula area = 1/2 (base*height) is true for regular triangles as well as for irregular triangles. Likewise, the formula area = width*height is true for regular rectangles (squares) as well as irregular rectangles.
Does this relationship continue for polygons with more corners? Is the formula for the area of a regular n-gon always the same as the formula for an irregular n-gon?
 A: $A = 1/2 b*h$ is not the formula for the area of a regular triangle.  It's an  formula for all triangles that can be applied inefficiently to a regular triangle.
This has two variables which are presumably given to us and has no indication how to find their values if they are not given to us.  In a regular triangle $h$ and $b$ are dependent upon each other so an efficient formula should only have one variable.
This is the formula for the area of a regular triangle:  $A = \sqrt{3}/2 * s^2$ where $s$ is the length of the side of the triangle.  We can do the same for the height: $A = h^2/\sqrt{3}$.
Likewise $A = b*h$ is not the formula for a regular rectangle.  A regular rectangle is a square and the formula for its area is $s^2$.  $A = bh$ is the area for a rectangle of which a square is a special case with $s = b = h$.  A rectangle is itself a special case of a trapezoid with the formula:  $A=1/2*(b_1 + b_2)*h$.  
The formula for a rectangle fails in applying to an irregular quadrilateral.  $A=b*h$?  Which base? How do we calculate height?  If we arbitrarily choose a base and the larger of the heights from the two remaining vertices,  the fourth vertex can be whereever it chooses and the area can bulge out and recess in and can't be determined by the other three vertices alone.  (In a rectangle or trapezoid we are restricted by the condition that the two bases are parallel.)
Which leads to the problem with your question.  A formula for a regular n-gon assumes equal sides and equal angles and will therefore only have two variables: $n$ the number of sides and some scaling factor, usually the side of the n-gon but sometimes it's referred via the radius, and sometimes the arithingummy-- the perpendicular from a side to the center.
With an irregular n-gon which sides, radii, altima etc. will you use to apply?
So the answer is no.  The question can't even be applied.
BUT the formula for any n-gon will be $\sum_{i=1}^n 1/2(s_i*h_i)$ where $s_i$ is the $i$th side and $h_i$ is the length of the perpendicular of the extended $s_i$ to an mutually arbitrary point inside the n-gon.  (i.e. $A = \text{ sum of the areas of triangles resulting from cutting the n-gon into n-triangles }$.)
The formula for a regular n-gon is simply a special application of that general formula with the assumption that all sides and angles (and thus the area of all triangles) are equal.
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Post-script:  In general a formula for an irregular n-gon will require $n-1$ variables; the first vertex can be fixed and the remaining $n-1$ vertexes supply remaining information.
In a triangle with 3 sides.  The fixed vertex,$v_1$ and the second vertex, $v_2$ determine the length of the base, $b$.  The third vertex, $v_3$, determines the height.
In a rectangle, $v_1$ and $v_2$ determine the base and $v_3$ determines the height.  $v_4$ is dependent on $v_1, v_2,$ and $v_3$ in the restriction that the sides must be perpendicular, so $v_4$ is superfluous.  In a trapezoid or an irregular quadrilateral it's position is necessary (although in trapezoid its vertical position isn't variable, only it lateral position).
For regular n-gons, the regularity assures that one's you have only two vertices determined, all remaining $n-2$ vertices are dependent entirely on those two.
