There are a number of ways to compute the area of a polygon from the coordinates of its vertices. There are algebraic formulas, geometric methods involving decomposing the figure, and an interesting method described by Pick's theorem.

<h3>Geometry</h3>

The figure can be enclosed by a rectangle that is 7 units high by 9 units wide. The actual figure is obtained by removing the triangular corners of this rectangle. The areas of those are given by the triangle area formula ...

A = 1/2bh

Working clockwise from lower left, the triangle base (horizontal) and height (vertical) dimensions are 5×6, 7×1, 2×6, 4×1. That means the area of the triangles is ...

A = 1/2(5×6 +7×1 +2×6 +4×1) = 1/2(30 +7 +12 +4) = 1/2(53) = 26.5

The area of the bounding rectangle is ...

A = bh = 9×7 = 63

So, the area of the figure is the difference ...

A = 63 -26.5 = 36.5 . . . square units

The other methods of finding the area have been added as attachments.

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2 years ago