Question

find the area of the polygon. square units

Explanation:

Step1: Divide the polygon

We can divide the given polygon into two triangles and a rectangle (or use the method of counting full squares and half - squares, or decompose it into simpler shapes). Let's decompose it into a small triangle at the top, a rectangle in the middle, and a large triangle at the bottom.

First, let's analyze the coordinates (assuming each grid square has a side length of 1 unit). Let's consider the base and height of each shape.

The small triangle at the top: Let's say its base is 2 units and height is 2 units. The area of a triangle is given by the formula $A=\frac{1}{2}\times base\times height$. So the area of the small triangle, $A_1=\frac{1}{2}\times2\times2 = 2$ square units.

Step2: Area of the rectangle

The rectangle in the middle: Let's assume its length is 2 units and width is 4 units. The area of a rectangle is $A = length\times width$. So the area of the rectangle, $A_2=2\times4 = 8$ square units.

Step3: Area of the large triangle

The large triangle at the bottom: Let's assume its base is 4 units and height is 4 units. Using the triangle area formula $A=\frac{1}{2}\times base\times height$, the area of the large triangle, $A_3=\frac{1}{2}\times4\times4=8$ square units.

Step4: Total area

Now, to find the total area of the polygon, we sum up the areas of the three shapes: $A = A_1+A_2+A_3=2 + 8+8=18$ square units.

(Another way: We can use the method of counting the number of full - grid squares and half - grid squares. Let's count the full squares and the half - squares. After careful counting, we find that the number of full squares and the equivalent of half - squares also gives us 18 square units. For example, some parts are triangles which are equivalent to half - squares when combined. )

Answer:

18