Lesson 4: Polygons

Property

A polygon is a closed, two-dimensional figure formed by three or more straight line segments that do not cross each other.

• Sides: The straight line segments that form the polygon.
• Vertices: The points where the sides meet (singular: vertex).
• Angles: The angles formed at the vertices inside the polygon.

Examples

Property

The area of a polygon on a coordinate plane can be found using two common methods:

1. Decomposition: The polygon is divided into smaller, non-overlapping shapes like triangles and rectangles. The total area is the sum of the areas of these smaller shapes.
   $$A_{polygon} = A_{shape1} + A_{shape2} + \dots$$
2. Enclosure (Box Method): A rectangle is drawn to enclose the entire polygon. The area of the polygon is the area of the rectangle minus the areas of the regions outside the polygon but inside the rectangle.
   $$A_{polygon} = A_{rectangle} - A_{outside\_regions}$$

Examples

• Decomposition Method: To find the area of the polygon, we can decompose it into a trapezoid and a smaller triangle. The trapezoid has area 9 square units, and the small triangle has area 3 square units. Adding them together, the total area is ($A = 9 + 3 = 12$) square units.

• Enclosure Method: To find the area of the polygon, we can enclose the polygon in a rectangle with area 20 square units, then subtract the areas of three right triangles outside the polygon, which are 6, 3, and 2.5 square units. The total area is ($A = 20 - (6 + 3 + 2.5) = 8.5$) square units.