WARNING: this message may contain more of a hint than you want.
If you want to work the last part out yourself, read only down
to the line of stars.

There are several ways to find the area of a convex polygon.  One way is
to draw enough interior horizontal lines in the figure to divide it into
trapezoids.  Another, traditional way, is to draw interior lines to divide
the area into triangles.

But there is a far simpler way.

Suppose the polygon is entirely above the x-axis (i.e., all y-values are
non-negative).  If P and Q are adjacent vertices of the polygon, let R be
the point on the x-axis directly below P, and let S be the point on the
x-axis directly below Q.  Then compute the area of the trapezoid PQSR.

For each line segment you get an area.  If P and Q are in the same
vertical line, that area is zero.  The area of the polygon is obtained
by adding or subtracting all those areas, according to the following rule.

If the vertices are listed in counter-clockwise order:
	The area of PQSR is added if Q is to left of P, and subtracted
	of Q is to the right of P.

If the vertices are listed in clockwise order:
	The area of PQSR is added if Q is to right of P, and subtracted
	of Q is to the left of P.

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 * THE ACTUAL FORMULA
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Suppose that V_0, V_1, .... V_n = V_0 are the vertices of the polygon,
in clockwise order.  Let V_i = (x_i,y_i).  Then the area A of the polygon
satisfies the equation:

	2*A =
(x_1-x_0)*(y_0+y_1) + (x_2-x_1)*(y_1+y_2) + ...  + (x_n-x_{n-1})*(y_{n-1}+y_n)

Each term will automatically be positive, negative, or zero, according
to the rule given above.  If the vertices are listed in counter-clockwise
order, then just reverse the sign.  It doesn't matter which vertex you
start with.  The "2" on the left-hand side comes from the fact that the
area of a trapezoid is the distance between the two parallel sides times
HALF the sum of the lengths of two parallel sides.

More generally, this formula is correct regardless of where the polygon
is placed (e.g., it could be entirely below, or even overlap, the x-axis),
and it works even for non-convex polygons.  (The definition of a polygon
requires that the boundary be a cycle of non-intersecting edges, but it
does not require that the interior be convex.)