Mathematics: Perimeter, Circumference, Area, and Volume

Introduction

• Overview of concepts: Perimeter, Circumference, Area, and Volume

Basic Definitions

• Length: One-dimensional measure (e.g., 1 cm)
• Area: Two-dimensional measure (e.g., 1 cm²)
• Volume: Three-dimensional measure (e.g., 1 cm³)

Units of Measurement

• Square Units: cm² (Area = cm * cm)
• Cubic Units: cm³ (Volume = cm * cm * cm)*

Perimeter

• Definition: Total distance around a shape
• Units: Length units (e.g., cm, inches, miles)
• Formula: Sum of all side lengths
• Example: Pentagon with each side 10 m -> Perimeter = 50 m
• Polygons: Sum of all side lengths
• Example 1: Right triangle with sides 3, 4, 5 cm -> Perimeter = 12 cm
• Example 2: Regular hexagon with side length 4 cm -> Perimeter = 24 cm
• Irregular polygon: Deduce missing sides, then sum all sides

Circumference and Area of Circles

• Circumference Formula: C = π * D
  • Diameter: Line through center of the circle
  • Memory Aid: Cherry Pies Delicious
• Area Formula: A = π * r²
  • Radius: Half of the diameter
  • Memory Aid: Apple Pies R²
• Example Calculation: Circle with radius 8 m
  • Circumference: 2 * 8 = 16; C = 3.14 * 16 = 50.24 m
  • Area: 8² = 64; A = 3.14 * 64 = 200.96 m²*

Area of Polygons

• Square/Rectangle: Area = length * width
  • Examples:
    • Square with side 2 cm -> Area = 4 cm²
    • Rectangle with length 4 cm and width 2 cm -> Area = 8 cm²
• Triangle: Area = ½ * base * height
  • Examples:
    • Acute triangle with base 5 m and height 8 m -> Area = 20 m²
    • Obtuse triangle with base 4 in and height 7 in -> Area = 14 in²
• Parallelogram: Area = base * height
  • Example: Base 5, height 2 -> Area = 10 cm²
• Trapezoid: Area = ½ * (base1 + base2) * height
  • Example: Bases 8 and 4, height 6 -> Area = 36 in²

Working with Complex Shapes

• Example Shape: Heart comprises square and semicircles
  • Square Area: Length 4, Width 4 -> Area = 16 cm²
  • Semicircle Area: Diameter 4, Radius 2 -> Full Circle Area = 12.56 cm²; Semicircle Area = 6.28 cm²
  • Total Area: 16 + 6.28 + 6.28 = 28.56 cm²

Volume

• Volume Calculation: Extend 2D shapes into 3D
• Square/Rectangular Prism: Volume = length * width * height
  • Example: Length 4, Width 3, Height 10 -> Volume = 120 cm³
• Triangular Prism: Volume = ½ * base * height * length
  • Example: Base 10, Height 8, Length 50 -> Volume = 2000 in³
• Cylinder: Volume = π * r² * height
  • Example: Radius 2, Height 10 -> Volume = 125.6 cm³*

Special Shapes

• Cone: Volume = ⅓ * π * r² * height
  • Example: Radius 5, Height 12 -> Volume = 314 in³
• Rectangular Pyramid: Volume = ⅓ * length * width * height
  • Example: Length 8, Width 7, Height 11 -> Volume = 205.33 in³
• Sphere: Volume = 4/3 * π * r³
  • Example: Radius 10 -> Volume = 4188.8 in³

Conclusion

• Emphasis on understanding key formulas and principles for TEAS exams
• Encouragement to reach out with questions and utilize additional resources for study