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Understanding Relationships Between Central Angles, Arc Lengths, and Sector Areas in Circles

darius-mccall avatar
darius-mccall
23 diapositives · 457 Views · Nov 18, 2014

Explore the relationships between central angles, arc lengths, and sector areas in circles through illustrated examples. Learn how to calculate the lengths of arcs and areas of sectors using given angles and radii, with step-by-step solutions provided. This guide includes practical exercises to reinforce your understanding, making abstract concepts relatable and easier to grasp. Perfect for students looking to improve their geometry skills and apply mathematical reasoning in real-world scenarios.

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Understanding Relationships Between Central Angles, Arc Lengths, and Sector Areas in Circles
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  1. CIRCLE

  2. Look at the picture RELATION BETWEEN THE CENTRAL ANGLE, SECTOR AREA, AND THE LENGHT OF ARCS BETWEEN TWO SECTORS B A 0

  3. B C Look at the picture  A  O D RELATION BETWEEN THE CENTRAL ANGLE, THE LENGHT OF ARCS, and SECTOR AREA

  4. B  O A If central angle is compared to a full angle of circle that is ( 3600), then :

  5. B  O A If the central angle that be compared to the full angle of circle ( 3600), then :

  6. B A 500 O 800 D C Example 1 : Look at the picture beside, the lenght of arc AB = 40 cm, AOB = 500, and AOB=800. Find the lenght of arc CD.

  7. B A 500 O 800 D C Solution : Given : AB = 40 cm, AOB = 500, AOB=800 Asking :Find the lenght of arc CD Answer: X = ( 40 x 80 ) : 50 = 64. So, we get the lenght of arc CD = 64 cm

  8. O 540 B A Example 2 : Look at the picture! The lenght of radii = 20 cm, AOB = 540. Determine the area of sector OAB

  9. Solution : Given : AOB = 540, and radii = 20 cm X = ( 3 x 1256 ) : 20 = 188,4 cm2. So, we get the area of sector OAB = 188,4 cm2.

  10. EXERCISE

  11. R O 450 P Q Question - 1 Look at the picture, the lenght of arc PQ = 50 cm, the lenght of arc QR = 75 cm and POQ = 450. Determine QOR.

  12. Solution : Given : The lenght of arc PQ = 50 cm The lenght of arc QR = 75 cm POQ = 450 X = ( 3 x 45) : 2 = 135 : 2 = 67,50 So, we get  QOR = 67,50.

  13. D B 600 O 12 cm A 12 cm C Question - 2 Look at the picture, COD = 600, the lenght of OA = 12 cm and AC = 12 cm. Calculate the area of the shaded region.

  14. Solution : Asking : the area of the shaded region. Given : Radii (1) = 12 cm Radii (2) = 24 cm. AOB = 600 Answer: Area of sector OAB = 60/360 x area of circle = 1/6 x 3,14 x 12 x 12 = 3,14 x 24 = 75,36 cm2

  15. Area of sector OCD = 60/360 x area of circle = 1/6 x 3,14 x 24 x 24 = 3,14 x 96 = 301,44 cm2 The area of the shaded region: = area of sector OCD - area of sector OAB = 301,44 cm2 - 75,36 cm2 = 226,08 cm2.

  16. 42 cm Question - 3 Calculate the area of the shaded region as shown in the figure above !

  17. 42 cm Solution: The area of circle that is shaded : L = ½  r2 = ½ x 22/7 x 21 x 21 = ½ x 22 x 63 = 11 x 63 = 693 cm2 The area of small circle that is shaded = The area of small circle that isn’t shaded.

  18. QUIZ

  19. C O 1200 400 A B Quiz - 1 Look at the picture, the area of sector OAB =60 cm2, AOB = 400 and BOC = 120o Find the area of sector OBC.

  20. 14 cm Quiz - 2 Calculate the area of the shaded region as shown in the figure above !

  21. Solution Quiz 1 : Given : AOB = 400 and BOC = 1200 The area of sector OAB = 60 cm2 Asking: The area of sector OBC. Answer: X = 3 x 60 = 180 cm2. So we get the area of sector OBC = 180 cm2.

  22. 14 cm Solution Quiz 2: The area of circle that is shaded : Lb =  r2 = 22/7 x 7 x 7 = 154 cm2 Lk =  r2 = 22/7 x 3,5 x 3,5 = 38,5 cm2 The area of the shaded region = 154 - 38,5 = 115,5 cm2

  23. I Like Mathematics