Pei Hwa Arc Length, Sector Area and Radian Measure Practice 1

Pei Hwa Secondary School Mathematics Department Secondary 4 NA Review Worksheet Name: ______________________________________ Class: ________ TG: ______ Date: ____________ Chapter 2: Arc Length, Sector Area and Radian Measure Success Criteria Success Criteria Indicate confidence level for each success criteria. 2.1 Length of Arc 1. I can identify what is arc length. 2. I can find arc length with given angle in degrees. 3. I can find angle with given arc length. 4. I can solve word problem involving arc length. 2.2 Area of sector 5. I can identify what is a sector of a circle. 6. I can find sector area with given angle in degrees. 7. I can find radius with given area sector. 8. I can identify what is a segment of a circle. 9. I can find area of segment. 10. I can solve word problem involving sector area and segment. 2.3 Radian measure 11. I can convert radians to degree. 12. I can convert degree to radian. 13. I can use calculator to evaluate trigonometric ratio of angles. 14. I can use calculator to evaluate inverse trigonometric ratio. 15. I can solve problems involving radian measure. 2.4 Arc length and sector area using radian measure 16. I can find arc length with given angle in radian measure. 17. I can find area sector with given angle in radian measure.

18. I can find segment with given angle in radian measure. 19. I can solve word problems involving radian measure.

1. Find the area of a sector of a circle with angle 5 4  radians and radius 9 cm. [2023, Dunearn Sec, Prelims, P1, Q6] 2. In the diagram, A and B lie on the circumference of a circle and O is the centre of the circle Given that angle AOB = 2.16 radians and the radius of the circle is 44 cm, calculate the perimeter of the minor sector, AOB. [2022, Kranji Sec, MYE, P2, Q4] 3. P and Q are points on the circle centre O. Angle POQ = 2.4 radians. The radius of the circle is 7 cm. Calculate (i) the length of minor arc PQ, (ii) the area of the segment shaded in the diagram. [2023, Bukit View Sec, Prelims, P2, Q11b] 4. The diagram shows a sector of a circle with radius 9 cm and angle x radians.

The sector has a perimeter of 31.2 cm. Calculate x . [2023, Hillgrove Sec, Prelims, P1, Q16] 5. A circle has centre O and radius 8 cm. The reflex angle 200 = AOC . (a) Calculate the minor arc length ABC. (b) Calculate the area of the minor shaded segment. [2023, Bendemeer Sec, Prelims, P1, Q22] 6. The points A and B lie on a circle centre O. The radius of the circle is 8 cm and ∠𝐴𝑂𝐵 is 1.72 radian. Find the (i) length of minor arc AB, (ii) length of line AB, (iii) perimeter of the shaded region. [2018, Boon Lay Sec, Prelims, P2, Q12a] 7. In the diagram, O is the centre of the circle. 1.85AOB= radians and OB = 8cm. Calculate (i) the length of the chord AB, (ii) the area of minor segment ABC. [2018, Fuhua Sec, Prelims, P2, Q12b]

8. In the diagram, AB and PQ are arcs of a circle, centre O, with radii 8 cm and 5 cm respectively. Arc PQ has a length of 6.5 cm. (a) Find angle AOB. (b) Find the perimeter of ABPQ. [2023, Queensway Sec, Prelims, P1, Q19] 9. In the diagram, OAB is a sector of a circle, centre O and radius 15 cm. C lies on OA such that AC = 4 cm. 1.4 rad.AOB= Find the area of the shaded region. [2023, Hua Yi Sec, Prelims, P1, Q18] 10. In the diagram, a toy-car wheel of radius 5 cm is in contact with the horizontal ground at P and touches the stair at R. The length of PQ = 3 cm. (i) Show that angle POR is approximately 0.644 rad. (ii) Calculate the area of sector POR. (iii) Calculate the area of the shaded region. [2018, Naval Base Sec, MYE, P2, Q12b]

11. In the diagram, P, Q, S are points on the circle with centre O and radius, r cm. OPT is a right-angled triangle, RPT is a straight line, ST = 8 cm and PT = 12 cm. (a) By using Pythagoras’ Theorem or otherwise, show that 5r = cm. (b) Calculate angle POT . (c) Find the length of major arc PQS. (d) Find the area of the shaded segment. [2019, Geylang Methodist Sec, Prelims, P2, Q9]

Answer Key: 1. 2159 cm 2. 183.04 cm 3. (i) 16.8 cm (ii) 242.3 cm 4. 7115 5. (a) 22.3 cm (b) 278.4 cm 6. (i) 13.76 cm (ii) 12.1 cm (iii) 25.9 cm (3 s.f) 7. (a) 12.8 cm (b) 228.4 cm 8. (a) 74.5 (b) 22.9 cm 9. 276.2 cm 10. (b) 28.05 cm (c) 20.95 cm 11. (b) 67.4 (c) 25.5 cm (d) 23.16 cm

Worked Solutions: 1. ( ) 2 2 15area of sector 9 24 159 cm =    = 2. perimeter 2.16 44 2(44) 183.04 cm =  + = 3. (i) minor arc 7 2.4 16.8 cm = = (ii) 22 2 11shaded segment 7 2.4 7 sin 2.422 42.3 cm    =   −         = 4. arc length 31.2 2(9) 13.2 cm r 13.2 9 13.2 7115 x x x =− = = = = 5. (a) ( )360 200minor arc 2 8 360 22.3 cm − =  = (b) ( ) ( ) ( ) 22 2 360 200 1shaded segment 8 8 sin 360 200360 2 78.4 cm − =  − −  = 6. (i) arc 8 1.72 13.76 cm AB= = (ii) 2 2 2 22 8 8 2(8)(8)cos1.72 8 8 2(8)(8)cos1.72 12.125 12.1 cm AB AB = + − = + − = = (iii) perimeter of shaded region 13.76 12.125 25.9 cm (3 s.f) =+ = 7. (a) 2 2 2 22 8 8 2(8)(8)cos1.85 8 8 2(8)(8)cos1.85 12.777 12.8 cm AB AB = + − = + − = = (b) ( ) ( ) ( ) 22 2 11shaded segment 8 1.85 8 sin 1.8522 28.4 cm =   − =

8. (a) Arc 6.5 2 5 6.5360 6.5 360 2 (5) 74.484 74.5 PQ x x AOB   =   = = = = (b) 74.484perimeter 2 (8) 2(3)360 22.9 cm =  + = 9. ( ) ( ) ( )( ) 2 2 2 1area sector 15 1.4 2 157.5 cm 15 4 11 cm 1area of shaded region 157.5 15 11 sin1.42 76.2 cm OAB OC = = =− = =− = 10. (a)   1 3sin 5 3sin 5 0.644 rad shown POR POR − = =  = (b) 2 2 1area of sector 5 0.6442 8.05 cm =   = (c) ( ) 22 2 2 5 5 3 1 cm 1area of trapezium 5 1 32 9 cm area of shaded region 9 8.05 0.95 cm RQ= − − = =  +  = =− = 11. (a) ( )   2 2 2 2 22 22 8 12 16 64 144 16 144 64 5 cm shown OT OP PT rr r r r r r =+ + = + + + = + =− =

(b) 1 12tan 5 12tan 5 67.3801 67.4 POT POT − = =  = = (c) reflex 360 67.3801 360 67.3801length of major arc 2 (5) 360 25.5 cm POS PQS   = −  − =  = (d) ( ) ( ) 22 2 67.3801 1area of segment 5 5 sin 67.3801360 2 3.16 cm =  −    =