Geometry Lecture: Calculating Areas of Various Shapes

1. Rectangle

• Formula: Area = Length × Width
• Example: Length = 8, Width = 5
  • Area = 8 × 5 = 40 square units
  • Units are important: If given in feet, result is square feet.

2. Triangle

Right Triangle

• Formula: Area = 1/2 × Base × Height
• Example: Base = 10, Height = 8
  • Area = 1/2 × 10 × 8 = 40 square units

Non-right Triangle

• Example: Base = 11 (sum of 5 and 6), Height = 9
  • Area = 1/2 × 9 × 11 = 49.5 square units

Equilateral Triangle

• Formula: Area = (√3/4) × side²
• Example: Side = 10
  • Area = (√3/4) × 10² = 25√3 square units

3. Square

• Formula: Area = side²
• Example: Side = 9
  • Area = 9² = 81 square units

4. Circle

• Formula: Area = π × radius²
• Given Diameter: Diameter = 10
  • Radius = 5
  • Area = π × 5² = 25π square cm

Sector of a Circle

• Formula: Area = (θ/360) × π × radius²
• Example: Radius = 10, θ = 60°
  • Area = (60/360) × π × 10² = (50π/3) square cm

Semicircle

• Formula: Area = 1/2 × π × radius²
• Example: Radius = 8
  • Area = 32π square units

5. Parallelogram

• Formula: Area = Base × Height
• Example: Base = 8, Height = 12
  • Area = 8 × 12 = 96 square units

With Slant Height

• Find Height Using: a² + b² = c² (Pythagorean theorem)
• Example: Base = 9, Slant Height = 5, Segment = 3
  • Height = 4, Area = 9 × 4 = 36 square units

6. Trapezoid

• Formula: Area = 1/2 × (Base1 + Base2) × Height
• Example: Base1 = 10, Base2 = 20, Height = 8
  • Area = 1/2 × (10 + 20) × 8 = 120 square units

Complex Trapezoid

• Divide into Right Triangles
• Calculate Height Using Pythagorean Theorem
• Example: Base1 = 12, Base2 = 24, Height from Triangle = 8
  • Area = 1/2 × (12 + 24) × 8 = 144 square units

7. Rhombus

• Formula: Area = 1/2 × Diagonal1 × Diagonal2
• Example: Diagonal1 = 10, Diagonal2 = 12
  • Area = 1/2 × 10 × 12 = 60 square units

With Right Triangle

• Identify Special Triangles (e.g. 5-12-13)
• Example: Side = 13, Half Diagonal = 5
  • Diagonal1 = 10, Diagonal2 = 24
  • Area = 1/2 × 10 × 24 = 120 square units

8. Triangle with Known Sides and Angle

• Formula: Area = 1/2 × a × b × sin(C)
• Example: Sides = 12, 10; Angle = 30°
  • Area = 30 square units

9. Heron's Formula for Scalene Triangle

• Formula:
  • s = 1/2 Perimeter
  • Area = √[s × (s-a) × (s-b) × (s-c)]
• Example: Sides = 9, 10, 11
  • s = 15, Area = 30√2 square units

10. Square with Given Diagonal

• Find Side Using Pythagorean Theorem
• Example: Diagonal = 10√2
  • Side = 10, Area = 100 square units

11. Shaded Region in Circle

• Formula: Area of Circle - Area of Triangle
• Example: Radius = Base = Height = 8
  • Area = 64π - 32 = 64π - 32 square units