Level: Board

This Course consists of CBSE Class 10 Mathematics Study Material.

Notes on Area of a Polygon Compared to a Flat Surface

1. Definition of Area in a Polygon

• The area of a polygon refers to the amount of space enclosed within its boundaries.
• It is measured in square units (e.g., square meters, square centimeters).

2. Comparison with a Flat Surface

• A flat surface (e.g., a plane) is an infinitely extending, two-dimensional space with no boundaries.
• A polygon is a bounded portion of a flat surface, meaning it has a finite area.

3. Key Differences

4. Formulas for Polygon Area

• Triangle:
  $A = \frac{1}{2} \times \text{base} \times \text{height}$ 
• Rectangle:
  $A = \text{length} \times \text{width}$ 

  $A=\text{length}\times \text{width}$

• Square:
  $A = \text{side}^2$ 
• Regular Polygon (n sides):
  $$$$ 

   

  $$A = \frac{1}{4} n s^2 \cot\left(\frac{\pi}{n}\right)$$ 

  where, $s$ is the side length, and $n$ is the number of sides.

• Irregular Polygons: Area is found using the Shoelace Theorem or by dividing into simpler shapes.

5. Special Cases

• Circle (not a polygon):
  $A = \pi r^2$ 

  (since it has no straight edges).

• Curved Shapes: Require calculus for precise area computation.

6. Conclusion

• A polygon is a subset of a flat surface, meaning its area is always finite.
• The total area depends on the number of sides and dimensions of the polygon.
• While a flat surface is infinite, any polygon represents a measurable portion of it.

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