To find the area and perimeter of the polygon in the image, we first identify the coordinates of the points labeled $A$, $B$, $C$, $D$, and $E$ based on the grid.

Coordinates of the points:

• $A(-2, 2)$
• $B(0, 4)$
• $C(4, 2)$
• $D(4, -2)$
• $E(-2, -2)$

Step 1: Calculate the Area

We can use the Shoelace Theorem (or Surveyor's Formula) to calculate the area of a polygon when the coordinates of its vertices are known.

The formula for the area is:

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + \dots + x_ny_1 - (y_1x_2 + y_2x_3 + \dots + y_nx_1) \right|$

Using the coordinates $A(-2, 2)$, $B(0, 4)$, $C(4, 2)$, $D(4, -2)$, $E(-2, -2)$, we substitute them into the formula:

$\text{Area} = \frac{1}{2} \left| (-2 \cdot 4) + (0 \cdot 2) + (4 \cdot -2) + (4 \cdot -2) + (-2 \cdot 2) - (2 \cdot 0) - (4 \cdot 4) - (2 \cdot 4) - (-2 \cdot -2) - (-2 \cdot -2) \right|$

Simplifying:

$\text{Area} = \frac{1}{2} \left| (-8) + (0) + (-8) + (-8) + (-4) - (0) - (16) - (8) - (4) - (4) \right|$ $\text{Area} = \frac{1}{2} \left| -28 \right| = 14 \text{ square units}$

Step 2: Calculate the Perimeter

The perimeter is the sum of the lengths of all the sides. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Now, let's calculate the distance between each pair of consecutive points:

1. Distance $AB$: $d_{AB} = \sqrt{(0 - (-2))^2 + (4 - 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

2. Distance $BC$: $d_{BC} = \sqrt{(4 - 0)^2 + (2 - 4)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$

3. Distance $CD$: $d_{CD} = \sqrt{(4 - 4)^2 + (-2 - 2)^2} = \sqrt{0 + (-4)^2} = \sqrt{16} = 4$

4. Distance $DE$: $d_{DE} = \sqrt{(-2 - 4)^2 + (-2 - (-2))^2} = \sqrt{(-6)^2 + 0} = \sqrt{36} = 6$

5. Distance $EA$: $d_{EA} = \sqrt{(-2 - (-2))^2 + (2 - (-2))^2} = \sqrt{0 + (4)^2} = \sqrt{16} = 4$

Now, sum all the distances to find the perimeter:

$\text{Perimeter} = 2\sqrt{2} + 2\sqrt{5} + 4 + 6 + 4$ $\text{Perimeter} = 14 + 2\sqrt{2} + 2\sqrt{5}$

You can either leave the perimeter in this simplified radical form or calculate the approximate decimal value:

$\text{Perimeter} \approx 14 + 2(1.414) + 2(2.236) = 14 + 2.828 + 4.472 \approx 21.3 \text{ units}$

Final Answers:

• Area: $14$ square units
• Perimeter: $14 + 2\sqrt{2} + 2\sqrt{5}$ or approximately $21.3$ units

Would you like more details on any step or have any questions? Here are some follow-up questions:

1. How does the Shoelace Theorem work in three-dimensional space?
2. How do you convert a polygon's area formula into integrals?
3. What are some alternative methods for calculating the perimeter of polygons?
4. Can the area of irregular polygons always be calculated using the Shoelace Theorem?
5. How does the distance formula generalize to higher dimensions?

Tip: For polygons on a