Calculating the Area of Polygons
Calculating the Area of Polygons teaches Grade 6 students two methods for finding polygon area on a coordinate plane: decomposition (splitting into triangles and rectangles) and the enclosure method (finding a surrounding rectangle area and subtracting surrounding shapes). Covered in Illustrative Mathematics Grade 6, Unit 1: Area and Surface Area, both methods rely on the coordinate positions of vertices to compute areas of simpler shapes that together make up the polygon.
Key Concepts
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.
Common Questions
How do you find the area of a polygon on a coordinate plane?
Either decompose the polygon into triangles and rectangles whose areas you can calculate, or enclose it in a rectangle and subtract the areas of the shapes outside the polygon.
What is the decomposition method for polygon area?
Divide the polygon into non-overlapping triangles and rectangles. Calculate each area separately and add them together.
What is the enclosure method for polygon area?
Draw the smallest rectangle that contains the polygon. Calculate the rectangle area, then subtract the areas of the right triangles (and other shapes) that are inside the rectangle but outside the polygon.
Where is calculating the area of polygons in Illustrative Mathematics Grade 6?
This topic is in Unit 1: Area and Surface Area of Illustrative Mathematics Grade 6.
Why are coordinate grid polygons useful for area practice?
The grid makes it easy to measure horizontal and vertical distances, calculate rectangle and triangle areas, and decompose shapes systematically.