Chapter 11.5 - Area of Regular Polygons

Area of Regular Polygons

• Definition of Regular Polygons: A polygon with all sides and all angles congruent.

Equilateral Triangle

• Area Formula: Area = ( \frac{1}{2} \times base \times height )

• Example: For an equilateral triangle with side length (8):

  • Base = (8)

  • To find height, draw height (h) splitting triangle into 30-60-90 triangles.

  • Angles: 60°, split into two 30° angles.

    • Opposite side of 30°= (\frac{base}{2} = \frac{8}{2} = 4)

    • Height (h) = side length × ( \frac{\sqrt{3}}{2} = 4\sqrt{3})

• Area Calculation:

  • Substituting values:

  • Area = ( \frac{1}{2} \times 8 \times 4 \sqrt{3} = 16\sqrt{3} ) square units.

General Formula for Equilateral Triangle

• If side length is (s):

  • Height = ( \frac{s}{2} \times \sqrt{3})

  • General Area Formula: ( \text{Area} = \frac{s^2 \sqrt{3}}{4} )

  • Reminder: Mnemonic to remember: 2, 3, 4

Area of a Regular Hexagon

• Lack of a specific formula prompts splitting the hexagon into triangles (6 equilateral triangles).

  • Side length of hexagon: (18) meters.

  • Each triangle's area = Area of equilateral triangle.

  • Area Calculation:

    • Area = ( 6 \times \frac{s^2 \sqrt{3}}{4} )

    • Using side length (18): Area = ( 486\sqrt{3} ) square meters.

General Strategy for Regular Polygons

• Understanding the Center:

  • Radius: Line from center to any vertex (All radii are congruent).

  • Apothem: Line from center to the midpoint of a side (All apothems are congruent).

• Properties:

  • An apothem is perpendicular to a side, and also bisects the central angle formed by two radii.

Area Formula for Regular Polygons

• Area = ( \frac{1}{2} \times perimeter \times apothem )

• Example Applying this Formula:

  • Regular hexagon perimeter: (72) cm, side length: (12) cm, find apothem using triangle properties.

  • Area = ( \frac{1}{2} \times P \times A )

  • Area of hexagon re-calculated and finds equivalently using perimeter and apothem.

Key Takeaways

• Remember the formulas for specific regular polygons:

  • Equilateral Triangle: ( \frac{s^2 \sqrt{3}}{4} )

  • Regular Polygons in general: ( \frac{1}{2} \times perimeter \times apothem )