Calculating the Area of a Regular Polygon

Key Concepts

• The area of any regular polygon can be calculated using the formula:

  [ \text{Area} = \frac{1}{2} \times \text{apothem} \times \text{perimeter} ]

• Apothem: A line from the center to the midpoint of one of its sides.

• Perimeter: The total length of all the sides of the polygon.

Example Calculations

Regular Hexagon

• Given:
  • Apothem ( a = 8\sqrt{3} )
  • Side length ( s = 16 )
• Steps:
  1. Calculate the perimeter:
     • Perimeter ( = 6 \times 16 = 96 )
  2. Calculate the area:
     • Area ( = \frac{1}{2} \times 8\sqrt{3} \times 96 = 384\sqrt{3} )

Regular Pentagon

• Given:
  • Apothem ( a = 9 )
  • Side length ( s = 10.6 )
• Steps:
  1. Calculate the perimeter:
     • Perimeter ( = 5 \times 10.6 = 53 )
  2. Calculate the area:
     • Area ( = \frac{1}{2} \times 9 \times 53 = 238.5 )

Regular Hexagon with Side Length

• Given:
  • Side length ( s = 20 )
• Steps:
  1. Calculate the angle ( \theta = \frac{360}{2n} ); ( n = 6 )
     • ( \theta = 30^\circ )
  2. Use the 30-60-90 triangle properties:
     • Apothem ( a = 10\sqrt{3} )
  3. Calculate the perimeter:
     • Perimeter ( = 6 \times 20 = 120 )
  4. Calculate the area:
     • Area ( = \frac{1}{2} \times 10\sqrt{3} \times 120 = 600\sqrt{3} )

Regular Pentagon with Only Side Length

• Given:
  • Side length ( s = 30 )
• Steps:
  1. Calculate the angle ( \theta = \frac{360}{2n} ); ( n = 5 )
     • ( \theta = 36^\circ )
  2. Calculate the apothem using trigonometry:
     • Apothem ( a \approx 20.65 )
  3. Calculate the perimeter:
     • Perimeter ( = 5 \times 30 = 150 )
  4. Calculate the area:
     • Area ( = \frac{1}{2} \times 20.65 \times 150 \approx 1550 )

Equilateral Triangle

• Given:
  • Side length ( s = 20 )
• Steps:
  1. Use the formula for equilateral triangles:
     • Area ( = \frac{\sqrt{3}}{4} \times 20^2 = 100\sqrt{3} )

Regular Hexagon with Given Radius

• Given:
  • Radius ( r = 10 )
• Steps:
  1. Calculate the angle and use 30-60-90 triangle:
     • Apothem ( a = 5\sqrt{3} )
     • Side ( s = 10 )
  2. Calculate the perimeter:
     • Perimeter ( = 6 \times 10 = 60 )
  3. Calculate the area:
     • Area ( = \frac{1}{2} \times 5\sqrt{3} \times 60 = 150\sqrt{3} )

Summary

• You can calculate the area of any regular polygon using different known values such as side length, apothem, or radius.
• Utilize trigonometric functions and properties of special triangles (30-60-90) to find missing dimensions when necessary.

Note: Practice these steps with various exercises to become proficient in calculating areas of polygons.