(3) Area Regular Polygons

Introduction to Area of Regular Polygons

• Concept of finding areas of regular polygons by breaking them into simpler shapes, specifically triangles.

Area of a Regular Hexagon with Side Length of 8 cm

• A regular hexagon is composed of 6 equilateral triangles.

• Key Formula: Area of equilateral triangle = (base × height) / 2.

• Dividing the hexagon, each triangle has a side length of 8 cm.

Finding the Area of One Equilateral Triangle

• Triangle Setup:

  • Base = 8 cm,

  • Cut triangle into half, forming a right triangle:

    • One leg = 4 cm (half of base),

    • Hypotenuse = 8 cm.

• Use Pythagorean Theorem to find height:

  • Formula: a² + 4² = 8².

  • Calculation: a² + 16 = 64

    • a² = 64 - 16 = 48

    • a = √48.

• Height: √48 cm.

• Area of one triangle: (base × height) / 2

  • = (8 × √48) / 2

  • = 4 × √48 cm².

Total Area of Hexagon

• Total Area = 6 × Area of one triangle

  • = 6 × (4 × √48)

  • = 24 × √48 cm².

• Approximation (via calculator):

  • Approximately 166.208 cm².

Area of a Regular Pentagon

• Pentagon can be divided into 5 congruent isosceles triangles.

• Each triangle's area calculated similarly:

  • Height: Called apothem, denoted as 'a'.

  • Example Situation:

    • Side length = 10 cm,

    • Apothem = 6.9 cm.

• Area Calculation:

  • Area = 5 × (base × height) / 2

  • = 5 × (10 × 6.9) / 2

  • Final Approximation: Approximately 172.5 square inches.

Area of Other Regular Polygons: Heptagons and n-gons

• For heptagon, again expressed through the area of 7 triangles.

• Formula generalization:

  • Area = (n × side length × apothem) / 2,

  • Where n is the number of sides.

General Formula for Area of Regular Polygons

• Conjecture: Area = 1/2 × A × S × N,

  • A = apothem,

  • S = side length,

  • N = number of sides.

• Alternate Expression: Area = 1/2 × apothem × perimeter (P = side length × N).

Example Problems for Practice

1. Pentagon Problem: Given side length 107.5 cm, and area ~19,886.5 cm², find approximate apothem.

2. Area Calculation: Side length = 24 cm, apothem ~24.9 cm, calculate area.

3. Dodecagon: What is area if perimeter is approximately 81.6 cm?

4. Nonagon Problem: Side length of nonagon with apothem of 9 in, area ~259.210 square inches.