Avoid These Errors: Calculating Area of Decomposed Polygons (Grade 6)

📚 Understanding Decomposed Polygons

Decomposed polygons are shapes formed by combining or dividing simpler polygons, like rectangles, triangles, and squares. To find the total area, you calculate the area of each individual shape and then add them together.

📜 A Brief History

The concept of area calculation dates back to ancient civilizations like the Egyptians and Babylonians, who needed it for land surveying and construction. They developed basic formulas for calculating areas of simple shapes. The systematic approach to decomposing complex shapes builds upon these early foundations of geometry.

📐 Key Principles

• 📏 Decomposition: Break down the complex polygon into simpler shapes like rectangles, triangles, squares, and parallelograms.
• ➕ Area Calculation: Calculate the area of each of the simpler shapes using their respective formulas. For example, the area of a rectangle is length × width, and the area of a triangle is $\frac{1}{2}$ × base × height.
• ✅ Summation: Add the areas of all the simpler shapes to find the total area of the decomposed polygon.

➕ Real-World Examples

Example 1: A House Floor Plan

Imagine a floor plan that looks like a rectangle with a smaller rectangle attached to one side. To find the total area, you would:

1. Calculate the area of the larger rectangle.
2. Calculate the area of the smaller rectangle.
3. Add the two areas together.

Example 2: An Irregular Garden Bed

Suppose you have a garden bed that is shaped like a rectangle with a triangle on top. The procedure is very similar:

1. Find the area of the rectangular portion.
2. Find the area of the triangular portion.
3. Add both to arrive at the total area of the garden bed.

💡 Tips for Success

• 🔍 Visualize: Always draw or visualize the decomposed shapes clearly.
• 📝 Label: Label all the dimensions correctly to avoid confusion.
• ➗ Divide Carefully: Ensure shapes do not overlap and completely cover the original polygon.

🔢 Practice Quiz

Calculate the area of the following decomposed polygons:

1. A shape composed of a rectangle (length = 8 cm, width = 5 cm) and a triangle (base = 8 cm, height = 3 cm).
2. A shape composed of two rectangles: Rectangle A (length = 6 m, width = 4 m) and Rectangle B (length = 3 m, width = 2 m).
3. A shape composed of a square (side = 7 inches) and a triangle (base = 7 inches, height = 4 inches).
4. A shape composed of a rectangle (length = 10 ft, width = 6 ft) with a smaller rectangle cut out (length = 4 ft, width = 2 ft). (Hint: Subtract the cutout area).
5. A shape composed of two identical triangles, each with a base of 5 inches and a height of 6 inches.
6. A shape is made up of one square with sides of 4cm and two triangles of base 4cm and height 3cm.
7. A shape is composed of a rectangle of sides 6m and 4m, and a triangle of base 4m and height 2m.

Answers:

1. 64 cm²
2. 30 m²
3. 63 in²
4. 52 ft²
5. 30 in²
6. 22cm²
7. 28m²

✅ Conclusion

Calculating the area of decomposed polygons is a fundamental skill in geometry. By breaking down complex shapes into simpler ones, you can easily find their total area. Practice these techniques, and you'll master the art of area calculation in no time!