ಹಲಬದಿಗಳು – ಭಾಗ 2

ನಾವು ಹಿಂದಿನ ಬರಹದಲ್ಲಿ ಹಲಬದಿಗಳು ಎಂದರೇನು ಮತ್ತು ಹಲಬದಿಯ ಹಲವು ಬಗೆಗಳನ್ನು ತಿಳಿದುಕೊಂಡೆವು, ಈಗ ಹಲಬದಿಗಳ ಮೂಲೆ (Angle), ಸುತ್ತಳತೆ (Perimeter) ಮತ್ತು ಹರವನ್ನು (Area) ಕಂಡುಕೊಳ್ಳುವ ಬಗೆಗಳನ್ನು ತಿಳಿಯೋಣ ಬನ್ನಿ.

ಹಲಬದಿಯ ಸುತ್ತಳತೆ (Perimeter of a polygon):

ಯಾವುದೇ ಹಲಬದಿಯ ಸುತ್ತಳತೆಯನ್ನು ಅದರ ಎಲ್ಲಾ ಬದಿಗಳ ಉದ್ದವನ್ನು ಕೂಡಿಸಿ ಸುಲಭವಾಗಿ ಕಂಡುಹಿಡಿಯಬಹುದಾಗಿದೆ.

  • ಉದಾಹರಣೆಯಾಗಿ ಒಂದು ABCDE ಐದ್ಬದಿಯನ್ನು (Pentagon) ತೆಗೆದುಕೊಳ್ಳೋಣ.

Image1 P2ಐದ್ಬದಿಯ ಸುತ್ತಳತೆ P ಆಗಿರಲಿ, ಬದಿ1 = AB, ಬದಿ2 = BC, ಬದಿ3 = CD, ಬದಿ4 = DE, ಬದಿ5 = EA ಆದಾಗ

ಐದ್ಬದಿಯ ಸುತ್ತಳತೆ P = ಬದಿ1+ ಬದಿ2 + ಬದಿ3 + ಬದಿ4 + ಬದಿ5 = AB + BC + CD + DE + EA ಆಗಿರುತ್ತದೆ.

  • ಉದಾಹರಣೆಯಾಗಿ ಆರುಬದಿಯುಳ್ಳ ABCDEF ಎಂಬ ಒಂದು ತಗ್ಗು ಹಲಬದಿಯನ್ನು (Concave Polygon) ತೆಗೆದುಕೊಳ್ಳೋಣ.

Image2 P2ಆರುಬದಿಯುಳ್ಳ ABCDEF ಈ ತಗ್ಗು ಹಲಬದಿಯಲ್ಲಿ AB = 8cm, BC = 5cm, CD = 7cm, ED = 3cm, EF = 12cm, FA = 10cm ಆಗಿವೆ, ಸುತ್ತಳತೆ P ಆಗಿರಲಿ.

ಆರುಬದಿಯುಳ್ಳ ABCDEF  ಹಲಬದಿಯ ಸುತ್ತಳತೆ P = AB + BC + CD + DE + EF + FA = 8 + 5 + 7 + 3 + 12 + 10 = 45 cm ಆಗಿದೆ.

 

  • ಯಾವುದೇ ಹಲಬದಿಯ ಬದಿಗಳು n ಆದಾಗ ಅದರ ಸುತ್ತಳತೆ P = ಬದಿ1 + ಬದಿ2 + ಬದಿ3 + …+ …+ ಬದಿn-1 + ಬದಿn ಆಗಿರುತ್ತದೆ.

ಅದನ್ನು ಇನ್ನು ಸುಳುವಾಗಿ Image3 P2   ಎಂದು ಬರೆಯಬಹುದು, ಇಲ್ಲಿ i = 1,2,3……n, n ಎಂಬುವುದು ಹಲಬದಿಯು ಎಷ್ಟು ಬದಿಗಳನ್ನು ಹೊಂದಿದೆ ಎಂಬುವುದನ್ನು ತಿಳಿಸುತ್ತದೆ.

  • ಒಂದು ಸುಳುವಾದ ಹಲಬದಿಯು (Simple Polygon) ಸಾಟಿ ಹಲಬದಿಯಾದಾಗ (Regular Polygon) ಅದರ ಸುತ್ತಳತೆಯನ್ನು P= n x s = ಒಟ್ಟು ಬದಿಗಳು x ಒಂದು ಬದಿಯ ಅಳತೆ ಎಂದು ಬರೆಯಬಹುದು.

ಇಲ್ಲಿ n -> ಒಟ್ಟು ಬದಿಗಳು.

s -> ಒಂದು ಬದಿಯ ಅಳತೆ.

ಹಲಬದಿಯ ಒಳ ಮೂಲೆಗಳನ್ನು (Interior Angles)  ಕಂಡುಕೊಳ್ಳುವ ಬಗೆ:

  • ಯಾವುದೇ ಹಲಬದಿಯ ಒಳಮೂಲೆಗಳ ಮೊತ್ತವು (n − 2) π C ಆಗಿರುತ್ತದೆ,

ಇಲ್ಲಿ c ಗುರುತು ರೇಡಿಯನ್ಸ್ (Radians) ಅನ್ನು ತಿಳಿಸುತ್ತದೆ, ಒಂದು ರೇಡಿಯನ್ ಅನ್ನು 1c ಎಂದು ಬರೆಯಬಹುದು.

1 ನ ಬೆಲೆ 180°/π ಆಗಿರುತ್ತದೆ,

π C = 180° ಆಗಿದೆ,  ಇಲ್ಲಿ π = 3.14159 ಆಗಿದೆ.

ಹಲಬದಿಯ ಒಳಮೂಲೆಗಳ ಮೊತ್ತವನ್ನು (n − 2) × 180° ಎಂದು ಸುಳುವಾಗಿ ಬರೆದುಕೊಳ್ಳಬಹುದು.

ಇಲ್ಲಿ  n ಎಷ್ಟು ಬದಿಗಳನ್ನು ಹೊಂದಿವೆ ಎಂಬುವುದಾಗಿದೆ.

ಹಲಬದಿಯ ಒಳಮೂಲೆಗಳ ಮೊತ್ತ ವನ್ನು ತಿಳಿಸುವ ಸರಿಹೊಂದಿಕೆಯು (Equation) ಉಬ್ಬು ಹಲಬದಿ (Convex Polygon) ಮತ್ತು ತಗ್ಗು ಹಲಬದಿಗೂ (Concave Polygon) ಸರಿಹೊಂದುತ್ತುದೆ.

 

  • ಒಂದು ಹಲಬದಿಯು ಸಾಟಿ ಹಲಬದಿಯಾದಾಗ (Regular Polygon) ಅದರ ಯಾವುದೇ ಒಂದು ಮೂಲೆಯು 180° – 360°/n ಆಗಿರುತ್ತದೆ. ಇಲ್ಲಿ n ಎಷ್ಟು ಬದಿಗಳನ್ನು ಹೊಂದಿವೆ ಎಂಬುವುದಾಗಿದೆ.

 

ಉದಾಹರಣೆ1:  ಕೆಳಗೆ ಒಂದು ABCD ನಾಲ್ಬದಿಯನ್ನು (Quadrilateral) ಕೊಡಲಾಗಿದೆ, ಇದರ ಒಳಮೂಲೆಗಳ ಮೊತ್ತವೇನು?

Image4 P2ನಾವು ಮೇಲೆ ತಿಳಿದಂತೆ ಯಾವುದೇ ಹಲಬದಿಯ ಒಳಮೂಲೆಗಳ ಮೊತ್ತವು (n − 2) × 180° ಆಗಿದೆ,

ಒಂದು ನಾಲ್ಬದಿಯೆಂದರೆ ಅದು ನಾಲ್ಕು ಬದಿಗಳನ್ನು ಹೊಂದಿರುತ್ತದೆ, ಹಾಗಾಗಿ ಇಲ್ಲಿ n = 4 ಆಗುತ್ತದೆ.

ABCD ನಾಲ್ಬದಿಯ ಒಳಮೂಲೆಗಳ ಮೊತ್ತ BAD + ADC + DCB + CBA =  (n − 2) × 180° = (4 – 2) x 180° = 2 x 180 = 360° ಆಗಿದೆ.

ಉದಾಹರಣೆ2: ಕೆಳಗೆ ಒಂದು ಸಾಟಿ ಹನ್ನೆರಡುಬದಿಯನ್ನು(Regular Dodecagon) ಕೊಡಲಾಗಿದೆ, ಇದರ ಒಳ ಮೂಲೆಗಳ ಮೊತ್ತವೇನು ಮತ್ತು ಅದರ ಒಂದು ಮೂಲೆಯ ಬೆಲೆಯೇನು ?

 Image5 P2

ನಾವು ಮೇಲೆ ತಿಳಿದಂತೆ ಯಾವುದೇ ಹಲಬದಿಯ ಒಳಮೂಲೆಗಳ ಮೊತ್ತವು (n − 2) × 180° ಆಗಿದೆ,

ಮೇಲಿನ ಚಿತ್ರವು ಹನ್ನೆರಡು ಬದಿಯನ್ನು ಹೊಂದಿದ ಆಕಾರವಾಗಿದೆ, ಹಾಗಾಗಿ ಇಲ್ಲಿ n = 12 ಆಗುತ್ತದೆ.

ಸಾಟಿ ಹನ್ನೆರಡುಬದಿಯ ಒಳಮೂಲೆಗಳ ಮೊತ್ತ = (n − 2) × 180° = (12 – 2) x 180° = 1800° ಆಗಿದೆ.

ಈ ಹನ್ನೆರಡುಬದಿಯು(Dodecagon) ಒಂದು ಸಾಟಿ ಹಲಬದಿಯಾಗಿದೆ (Regular Polygon), ಅಂದರೆ ಅದರ ಎಲ್ಲಾ ಬದಿಗಳು ಒಂದೇ ಅಳತೆಯಲ್ಲಿರುತ್ತವೆ (Equilateral) ಹಾಗು ಅದರ ಎಲ್ಲಾ ಮೂಲೆಗಳು ಒಂದೇ ಅಳತೆಯಲ್ಲಿರುತ್ತವೆ (Equiangular).

ನಾವು ಮೇಲೆ ತಿಳಿದಂತೆ ಸಾಟಿ ಹಲಬದಿಯ (Regular Polygon) ಒಂದು ಮೂಲೆಯು 180° – 360°/n ಆಗಿರುತ್ತದೆ.

ಸಾಟಿ ಹನ್ನೆರಡುಬದಿಯ ಯಾವುದೇ ಒಂದು ಮೂಲೆ = 180° – 360°/n = 180°- 360°/12 = 180° – 30° = 150°ಆಗಿರುತ್ತದೆ.

 

ಉದಾಹರಣೆ 3: ಒಂದು ಸಾಟಿ ಹಲಬದಿಯ ಯಾವುದೇ ಮೂಲೆಯು 162° ಆಗಿದೆ, ಹಾಗಾದರೆ ಈ ಸಾಟಿ ಹಲಬದಿಯು ಎಷ್ಟು ಬದಿಗಳನ್ನು ಹೊಂದಿದೆ.

ಈ ಉದಾಹರಣೆಯಲ್ಲಿ ಸಾಟಿ ಹಲಬದಿಯ ಯಾವುದೇ ಮೂಲೆಯು 162° ಆಗಿದೆ.

ನಾವು ಮೇಲೆ ತಿಳಿದಂತೆ ಸಾಟಿ ಹಲಬದಿಯ (Regular Polygon) ಒಂದು ಮೂಲೆಯು 180° – 360°/n ಆಗಿರುತ್ತದೆ.

ಸಾಟಿ ಹಲಬದಿಯ ಒಂದು ಮೂಲೆ = 180° – 360°/n = 162°, ಇಲ್ಲಿ n ಎಂಬುವುದು ಅದರ ಒಟ್ಟು ಬದಿಗಳಾಗಿವೆ, ಇದನ್ನು ಕೆಳಗಿನಂತೆ ಬಿಡಿಸೋಣ

180° –  162° = 360°/n

18° = 360°/n

n = 360°/18 = 20

∴  ಸಾಟಿ ಹಲಬದಿಯ ಒಂದು ಮೂಲೆ 162° ಆದಾಗ ಅದು 20 ಬದಿಗಳನ್ನು ಹೊಂದಿರುತ್ತದೆ, ಈ ಸಾಟಿ ಹಲಬದಿಯನ್ನು ಇಪ್ಪತ್ತುಬದಿ (Icosagon)  ಆಕಾರ ಎಂದು ಕರೆಯಬಹುದು.

 Image6 P2

ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳುವ ಬಗೆ (Area of a Polygon):

ಮೂರ್ಬದಿ ಆಕಾರವು ಮೂರು ಬದಿಗಳನ್ನು ಹೊಂದಿರುತ್ತದೆ, ಆಯತ, ಚೌಕ ಇವುಗಳೆಲ್ಲವೂ ನಾಲ್ಕುಬದಿಗಳನ್ನು ಹೊಂದಿವೆ, ಹಾಗಾಗಿ ಇವುಗಳ ಹರವನ್ನು ಕಂಡುಹಿಡಿಯುವುದು ಸುಲಭ, ಆದರೆ ಇವುಗಳಿಗಿಂತ ಹೆಚ್ಚು ಬದಿ ಮತ್ತು ಬೇರೆ ಬೇರೆ ಅಳತೆಯ ಬದಿಗಳನ್ನು ಹೊಂದಿದ ಒಂದು ಸುಳುವಾದ ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಹಿಡಿಯುವುದು ಸುಲಭವಲ್ಲ. ಹಾಗಾಗಿ  ಚುಕ್ಕೆಗುರುತಿನ ಏರ್ಪಾಡನ್ನು (coordinate system)  ಬಳಸಿ ಎಲ್ಲಾ ಸುಳುವಾದ ಹಲಬದಿಗಳಿಗೆ ಒಗ್ಗುವಂತೆ ಹೀಗೆ ಬರೆಯಬಹುದು.

Image7 P2Image8 P2

  • n: ಬದಿಗಳು
  • x,y: ಹಲಬದಿಯ ತುದಿಗಳ ಚುಕ್ಕೆಗುರುತುಗಳು (Coordinates of polygon vertices)
  • k: 1, 2, 3, 4, …, n-1, n
  • ಇಲ್ಲಿ ಹರವು ಕಳೆಯುವ ಗುರುತನ್ನು (Negative Symbol) ಹೊಂದಿದ್ದರೆ ಅದನ್ನು ಕೂಡು ಗುರುತಿಗೆ(Positive Symbol) ಮಾರ್ಪಾಟು ಮಾಡಿಕೊಳ್ಳಬೇಕು, ಅದಕ್ಕೆ ದಿಟಬೆಲೆ ಗುರುತನ್ನು(absolute value/modulus/real number) ಬಳಸಬೇಕು,
  • ಉದಾಹರೆಣೆಗೆ -6 -> |6| -> 6, ಹಾಗಾಗಿ ಮೇಲಿನ ಸರಿಹೊಂದಿಕೆಗೆ ಗುರುತನ್ನು ಬಳಸಲಾಗಿದೆ

 ಕೇಳ್ವಿ 1:  ಒಂದು ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳಲು ಇಷ್ಟು ದೊಡ್ಡದಾದ ಸರಿಹೊಂದಿಕೆಯನ್ನು (equation) ಬಳಸಬೇಕೇ?

ಉತ್ತರ:  ಹೆಚ್ಚು ಹೆಚ್ಚು ಬದಿ ಮತ್ತು ಬೇರೆ ಬೇರೆ ಅಳತೆಯ ಬದಿಗಳನ್ನು ಹೊಂದಿದ ಒಂದು ಸುಳುವಾದ ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಹಿಡಿಯುವುದಕ್ಕೆ ಮೇಲಿನ ಸರಿಹೊಂದಿಕೆಯನ್ನು ಬಳಸಬಹುದು, ಮೂರ್ಬದಿ ಮತ್ತು ನಾಲ್ಬದಿ ಆಕಾರಗಳು ಕಡಿಮೆ ಬದಿಯನ್ನು ಹೊಂದಿದ್ದರಿಂದ ಅವುಗಳನ್ನು ಸುಲಭವಾಗಿ ಬೇರೆ ಬಗೆಯಾಗಿ ಅವುಗಳ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳಬಹುದು. ಮೂರ್ಬದಿಗಳ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳಲು ಈ ಬರಹವನ್ನು ಓದಿ ಮತ್ತು ನಾಲ್ಬದಿಗಳ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳಲು ಈ ಬರಹವನ್ನು ಓದಿ.

ಕೇಳ್ವಿ 2: ಒಂದು ಸುಳುವಾದ ಹಲಬದಿಯು (Simple Polygon) ಸಾಟಿ ಹಲಬದಿಯಾದಾಗ (Regular Polygon) ಮೇಲಿನ ಸರಿಹೊಂದಿಕೆಯನ್ನು ಬಳಸಿಕೊಳ್ಳಬೇಕೇ ?

ಉತ್ತರ: ಒಂದು ಸುಳುವಾದ ಹಲಬದಿಯು ಸಾಟಿ ಹಲಬದಿಯಾದಾಗ ಅದರ ಎಲ್ಲಾ ಬದಿಗಳು ಒಂದೇ ಅಳತೆಯನ್ನು ಹೊಂದಿರುತ್ತವೆ (Equilateral) ಮತ್ತು ಎಲ್ಲಾ ಮೂಲೆಗಳು ಒಂದೇ ಅಳತೆಯನ್ನು(Equiangular) ಹೊಂದಿರುತ್ತವೆ ಕೂಡ.

ಆ ಸಾಟಿ ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳಲು ಮೇಲಿನ ಸರಿಹೊಂದಿಕೆಯ ಬದಲು ಈ ಕೆಳಗಿನ ಸರಿಹೊಂದಿಕೆಯನ್ನು ಬಳಸಿಕೊಳ್ಳಬಹದು.

  • ಕೆಳಗಿನ ಚಿತ್ರದಲ್ಲಿ ತೋರಿಸಿರುವ ಎಲ್ಲಾ ಆಕಾರಗಳು ಸಾಟಿ ಹಲಬದಿಗಳಾಗಿವೆ (Regular Polygons).

Image9 P2

  • ಯಾವುದೇ ಒಂದು ಸಾಟಿ ಹಲಬದಿಯ ಹರವು A = 1/2 x (pa) ಆಗಿರುತ್ತದೆ.

ಇಲ್ಲಿ p à ಸುತ್ತಳತೆ (Perimeter)

       a à ನೇರಡ್ಡನಡುಗೆರೆ (Apothem)

ಹರವನ್ನು A = 1/2 x (pa) = 1/2 x (nsa) ಎಂದೂ ಬರೆಯಬಹುದು, ಏಕೆಂದರೆ ಸುತ್ತಳತೆ P= n x s =  ಒಟ್ಟು ಬದಿಗಳು x ಒಂದು ಬದಿಯ ಅಳತೆ ಆಗಿರುತ್ತದೆ.

ನೇರಡ್ಡನಡುಗೆರೆ (Apothem) ಎಂದರೆ ಒಂದು ಸಾಟಿ ಹಲಬದಿಯ ನಡುವಿಂದ ಅದರ ಒಂದು ಬದಿಗೆ ನೇರಡ್ಡವಾಗಿ ಎಳೆದ ಗೆರೆ.

  • ಉದಾಹರಣೆಯಾಗಿ  ಒಂದು ಸಾಟಿ ಎಂಟ್ಬದಿಯನ್ನು (Octagon) ತೆಗೆದುಕೊಂಡು ಸಾಟಿ ಹಲಬದಿಯ ಹರವಿನ ಸರಿಹೊಂದಿಕೆಯನ್ನು (Equation of area of regular polygon) ಕೆಳಗಿನಂತೆ ತೋರಿಸಬಹುದು.

Image10 P2

ಉದಾಹರಣೆ 1: ಸಾಟಿ ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳುವ ಸರಿಹೊಂದಿಕೆಯನ್ನು ಬಳಸಿ ಚೌಕದ ಹರವು ಬದಿ x ಬದಿ ಎಂದು ತೋರಿಸಿ.

Image11 P2ಚೌಕದಲ್ಲಿ ಎಲ್ಲಾ ನಾಲ್ಕು ಬದಿಗಳು (n=4) ಒಂದೇ ಅಳತೆಯನ್ನು ಹೊಂದಿರುತ್ತವೆ.

∴  ED = DG = GF = FE = s

ಚೌಕದ ನೇರಡ್ಡನಡುಗೆರೆಯ ಉದ್ದವು (length of apothem) ಚೌಕದ ಒಂದು ಬದಿಯ ಅರೆಪಾಲಿನ ಉದ್ದಕ್ಕೆ ಸರಿಯಾಗಿದೆ.

∴  ನೇರಡ್ಡನಡುಗೆರೆ a = s/2

ಸಾಟಿ ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳುವ ಸರಿಹೊಂದಿಕೆ

A = 1/2 x (pa) = 1/2 x (nsa) = 1/2 x (ಒಟ್ಟು ಬದಿಗಳು x ಬದಿಯ ಉದ್ದ x ನೇರಡ್ಡನಡುಗೆರೆಯ ಉದ್ದ)

∴  A = 1/2 x 4 x s x s/2 =2 x s x s/2 = s x s = ಬದಿ x ಬದಿ  ಆಗಿದೆ.

ಉದಾಹರಣೆ 2: ಒಂದು ಸಾಟಿ ಐದ್ಬದಿಯ (Regular Pentagon) ಬದಿಗಳು 7 cm  ಆಗಿವೆ ಮತ್ತು ಅದರ ನೇರಡ್ಡನಡುಗೆರೆ 4.81734 cm ಆದಾಗ ಅದರ ಹರವನ್ನು ಕಂಡುಹಿಡಿಯಿರಿ.

Image12 P2ಸಾಟಿ ಐದ್ಬದಿಯಲ್ಲಿ ಎಲ್ಲಾ ನಾಲ್ಕು ಬದಿಗಳು (n=5) ಒಂದೇ ಅಳತೆಯನ್ನು ಹೊಂದಿರುತ್ತವೆ.

ಸಾಟಿ ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳುವ ಸರಿಹೊಂದಿಕೆ.

A = 1/2 x (pa) = 1/2 x (nsa) = 1/2 x (ಒಟ್ಟು ಬದಿಗಳು x ಬದಿಯ ಉದ್ದ x ನೇರಡ್ಡನಡುಗೆರೆಯ ಉದ್ದ)

A = 1/2 x (5 x 7 x 4.81734) = 1/2 x (168.6069) = 84.30345 cm.

ಸಾಟಿ ಐದ್ಬದಿಯ ಹರವು A = 84.30345 cm.

ಉದಾಹರಣೆ 3: ಒಂದು ಸಾಟಿಯಲ್ಲದ ಹಲಬದಿ P1P2P3P4P5 ಯನ್ನು ಚುಕ್ಕೆಗುರುತಿನ ಏರ್ಪಾಟಿನಲ್ಲಿ (coordinate system) ಗುರುತಿಸಲಾಗಿದೆ, ಕೊಟ್ಟಿರುವ ಎಲ್ಲಾ ಗುರುತಿನ ಅಂಶಗಳನ್ನು ಬಳಸಿಕೊಂಡು ಈ ಹಲಬದಿಯ ಹರವನ್ನು ಕಂಡುಹಿಡಿಯಿರಿ.

Image13 P2

  • ಸಾಟಿಯಲ್ಲದ ಹಲಬದಿಯ ತುದಿಗಳನ್ನು(Vertices) ಚುಕ್ಕೆಗುರುತಿನ ಏರ್ಪಾಟಿನಲ್ಲಿ ಗುರುತಿಸಲಾಗಿದೆ
  • ಈ ಚುಕ್ಕೆಗುರುತುಗಳು(Coordinates) ಹೀಗಿವೆ P1(3,4), P2(5,11), P3(12,8), P4(9,5) ಮತ್ತು P5(5,6).
  • ಮೇಲೆ ಕೊಟ್ಟಿರುವ ಹಲಬದಿಯು ಸುಳುವಾದ ಹಲಬದಿಯಾಗಿದೆ (Simple Polygon) ಮತ್ತು ಸಾಟಿಯಲ್ಲದ ಹಲಬದಿಯಾಗಿದೆ (Irregular Polygon)
  • ಮೇಲೆ ತಿಳಿಸಿದಂತೆ ಸುಳುವಾದ ಹಲಬದಿಯ (Simple polygon) ಹರವನ್ನು ಕಂಡುಕೊಳ್ಳುವ ಸರಿಹೊಂದಿಕೆ (Equation).

Image14 P2 P = { P1(x1,y1),P2(x2,y2),P3(x3,y3),P4(x4,y4),P5(x5,y5)} = {P1(3,4),P2(5,11),P3(12,8),P4(9,5) P5(5,6)}  ಎಂದು ಹೊಂದಿಸಿಕೊಳ್ಳಬಹುದು.

Image15 P2ಹರವಿನ ಬೆಲೆ ಕಳೆಯುವ ಗುರುತನ್ನು (Negative Symbol) ಹೊಂದಿರುವುದರಿಂದ,  (Real/Absolute number symbol) ಗುರುತನ್ನು ಬಳಸಲಾಗಿದೆ.

 ಚಟುವಟಿಕೆ: ನಮ್ಮ ದಿನ ನಿತ್ಯದ ಬದುಕಿನಲ್ಲಿ ಕಾಣುವ ಎಲ್ಲಾ ಹಲಬದಿ ಆಕಾರಗಳನ್ನು ಗುರುತಿಸಿ, ಮತ್ತು ಅವುಗಳು ಯಾವ ಯಾವ ಬಗೆಯ ಹಲಬದಿಗಳಾಗಿವೆ ಎಂದು ಪಟ್ಟಿಮಾಡಿ (ಹಿಂದಿನ ಬರಹ  ಹಲಬದಿಗಳು ಭಾಗ 1 ನ್ನು ಬಳಸಿಕೊಳ್ಳಬಹುದು)

 (ಸೆಲೆಗಳು: dummies.com/education, easycalculation.com, math.blogoverflow.com, wikipedia.org)

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