Lesson 5: Composite and Irregular Shapes

Property

To find the area of a general polygonal figure, you can partition it into a combination of simpler shapes like rectangles and triangles, for which area formulas are known. The area of the original figure is the sum of the areas of the non-overlapping component figures.

Examples

• An L-shaped desk can be split into two rectangles: one $4 \times 2$ ft and another $5 \times 2$ ft. The total area is $(4 \times 2) + (5 \times 2) = 8 + 10 = 18$ square feet.
• A shape is made of a $6 \times 6$ cm square with a triangle on top. The triangle's base is 6 cm and height is 4 cm. The total area is $(6 \times 6) + (\frac{1}{2} \times 6 \times 4) = 36 + 12 = 48$ square cm.
• A polygon is composed of a central $10 \times 5$ rectangle and two identical triangles on each side, each with a base of 3 and height of 5. The area is $(10 \times 5) + 2 \times (\frac{1}{2} \times 3 \times 5) = 50 + 15 = 65$ square units.

Explanation

Don't have a formula for a weird shape? Just chop it up! By breaking a complex polygon into familiar pieces like rectangles and triangles, you can find the area of each piece and add them all up for the total.

Property

To find the area of a composite shape formed by removing a smaller rectangle from a larger one, subtract the area of the smaller rectangle from the area of the larger rectangle.

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.