11.1 : Understanding Area

Introduction to Area

• Chapter 11 focuses on the concept of area.
• Key question: What is area?

Definition of Area

• Area is the space covered by a two-dimensional shape.
• Important points:
  • Must be two-dimensional.
  • Can exist even without a specific formula.

Understanding Shapes and Area

• Irregular shapes also have area:
  • Example: A crazy shape can be analyzed by breaking it into known shapes, but it's often more effective to use a grid approach.
  • Counting square units in a grid can help approximate the area.
  • The more refined the grid, the more accurate the area becomes.
• If an exact area is needed for curved shapes, calculus is required.

Types of Shapes and Their Areas

• Enclosed Shapes: All enclosed shapes have area.
  • Non-enclosed shapes do not have area since there is uncertainty about their boundaries.

Congruent Shapes and Area

• Definition of congruent shapes:
  • Two shapes are congruent if they have
  • Corresponding sides equal (side, side, side property).
  • Corresponding angles equal.
• Theorem: If two shapes are congruent, they have the same area.

Reversibility of Area Congruency

• Not reversible: If two shapes have the same area, they do not necessarily have to be congruent.
  • Example: A triangle and rectangle may both have an area of 20 square units but are not congruent.

Example Problem: Calculating Area of a Shaded Region

• Scenario: A larger rectangle containing a smaller rectangle.
• Dimensions:
  • Large Rectangle: Length = 20 feet, Width = 10 feet
  • Small Rectangle: Length = 14 feet, Width = 6 feet
• Formula for area: Area = Length × Width

Steps to Calculate Shaded Area:

1. Calculate area of the larger rectangle:
   • Area = 20 × 10 = 200 square feet
2. Calculate area of the smaller rectangle:
   • Area = 14 × 6 = 84 square feet
3. Subtract the area of the smaller rectangle from the larger rectangle:
   • Shaded area = 200 - 84 = 116 square feet

Important Note on Units

• Always label your area in square units (e.g., square feet).
  • Area is expressed in units squared because it is calculated by multiplying two dimensions (feet × feet = feet²).