CIRCLE
RELATION BETWEEN THE CENTRAL ANGLE, SECTOR AREA, AND THE LENGHT OF ARCS BETWEEN TWO SECTORS Look at the picture
RELATION BETWEEN THE CENTRAL ANGLE, THE LENGHT OF ARCS, and SECTOR AREA Look at the picture central AOB = The lenght of arc AB The area of sector OAB central COD arc CD The area of sector OCD
Circumference of circle If central angle is compared to a full angle of circle that is ( 3600), then : O B A central AOB = The lenght of arc AB The area of sector OAB 3600 Circumference of circle Area of circle
If the central angle that be compared to the full angle of circle ( 3600), then : central AOB = The lenght of arc AB the area of sector OAB 3600 2r r2
Example 1 : O D C A B 500 800 Look at the picture beside, the lenght of arc AB = 40 cm, AOB = 500, and AOB = 800. Find the lenght of arc CD.
Solution : Given : AB = 40 cm, AOB = 500, AOB = 800 D C A B 500 800 Asking : Find the lenght of arc CD Answer: central AOB = The lenght of arc AB central COD The lenght of arc CD 500 = 40 cm 800 X cm X = ( 40 x 80 ) : 50 = 64. So, we get the lenght of arc CD = 64 cm
Example 2 : Look at the picture! The lenght of radii = 20 cm, AOB = 540. Determine the area of sector OAB O A B 540
Solution : Given : AOB = 540, and radii = 20 cm 540 = x 3600 r2 3 = central AOB = The area of sector OAB 3600 The area of circle 540 = x 3600 r2 3 = x 20 3,14 x 202 X = ( 3 x 1256 ) : 20 = 188,4 cm2. So, we get the area of sector OAB = 188,4 cm2.
EXERCISE
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. O P R 450 Q
Solution : 45 = 50 x 75 45 = 2 x 3 X = ( 3 x 45) : 2 = 135 : 2 = 67,50 Given : The lenght of arc PQ = 50 cm The lenght of arc QR = 75 cm POQ = 450 central POQ = The lenght of arc PQ central QOR The lenght of arc QR 45 = 50 x 75 45 = 2 x 3 X = ( 3 x 45) : 2 = 135 : 2 = 67,50 So, we get QOR = 67,50.
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. O C D B A 600 12 cm
Area of sector OAB = 60/360 x area of circle = 1/6 x 3,14 x 12 x 12 Solution : Given : Radii (1) = 12 cm Radii (2) = 24 cm. AOB = 600 Asking : the area of the shaded region. 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
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.
Question - 3 42 cm Calculate the area of the shaded region as shown in the figure above !
Solution : The area of circle that is shaded : L = ½ r2 42 cm 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.
QUIZ
Quiz - 1 Look at the picture, the area of sector OAB =60 cm2, 1200 400 B Look at the picture, the area of sector OAB =60 cm2, AOB = 400 and BOC = 120o Find the area of sector OBC.
Calculate the area of the shaded region as shown in the figure above ! Quiz - 2 14 cm Calculate the area of the shaded region as shown in the figure above !
Solution Quiz 1 : 400 = 60 1200 x 1 = 60 3 x X = 3 x 60 = 180 cm2. Given : AOB = 400 and BOC = 1200 The area of sector OAB = 60 cm2 Asking: The area of sector OBC. Answer: central AOB = The area of sector OAB central BOC The area of sector OBC 400 = 60 1200 x 1 = 60 3 x X = 3 x 60 = 180 cm2. So we get the area of sector OBC = 180 cm2.
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 14 cm The area of the shaded region = 154 - 38,5 = 115,5 cm2
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