AN fs) Circular measure> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK Before you start... ‘Cambridge Calculate the length Find the length of arc AB. IGCSE/O Level _| of an arc of a circle Mathematics | and the area of a sector of a circle, b Find the area of sector OB. 8.1 Circular measure ‘Have you ever wondered why there are 360. ‘degrees in one complete revolution? ‘The original reason for choosing the degree as a unit of angular measure is not known, but there are a number of different theories: ‘* ancient astronomers claimed that the Sun advanced in its path by one degree each day and that a solar year consisted of 360 days * the ancient Babylonians divided the circle into 6 equilateral triangles and then subdivided each angle at O into 60 further parts, resulting in 360 divisions in one complete revolution * 360 has many factors which makes division of the circle so much easier. Degrees are not the only way in which you can measure angles. In this section you will learn how to use radian measure. This is sometimes referred to as the natural unit of angular measure and it is used extensively in mathematics because it can simplify many formulae and calculations. In the diagram, the magnitude of angle AOB is 1 radian (1 radian is written as Trad or I), Anare equal in length to the radius of a circle subtends an angle of 1 radian at the centre, {It follows that the circumference (an are of length 27r) subtends an angle of 2 radians at the centre, 2m radians = 360° rradians = 180° When an angle is written in terms of a, the radian symbol is usually omitted. ‘Hence, ar = 180°, 176 >8 Circular measure Converting from degrees to radians Since 180° = m, then 90° = 5, 45° =F, ete, ‘Angles that are not simple fractions of 180° can be converted using the following rule: To convert from degrees to radians, multiply by +5 Converting from radians to degrees Since 1 = 180°, § = 30°, 75 = 18" etc. Angles that are not simple fractions of 7 can be converted using the following rule: To change from radians to degrees, multiply by ®°, (tis useful to remember that I radian = 1 x 182 = 57%) oes ees a Change ne to radians, giving your answer in terms of mr. b Change 3 radians to degrees. Answers 2 Method t: ‘Method 2: 180° = radians (60 735) radians 180 180, (42)' =F raaians 60° = 5 radians b Method 1: ‘rradians = 180° radians = 36° F radians = 36 an face F radians = 108" Exercise 8.1 1 Convert these angles to radians, in terms of =. a 10° b 20° < 40° d 50° e 1s f 1P 0g 135" oh 25 i360" jf 720" k 80° 13007 om n 75° 210° ——SSEE———— _ —————— SSE 17>> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 2 Convert these angles to degrees z z = = 2m aig ei 6 on a 9 3 4a In Sa ; > os nr FS, 9 0 On ' 0 io ox Tr 8 on ck & ee nm ® o & °& 3. Write each of these angles in radians, correct to 3 sf. a a b 5° ce 34 a 123° e uP 4 Write each of these angles in degrees correct to 1 decimal place. a 13rd b 25rad ¢ 102rmd d 183rad e O.S8rad 5 Copy and complete the tables, giving your answers in terms of a © [5 [50 [135] 180] 225] 270] 515] 30] 0 7 2a] & es 0 [0] a 150] 180 “fee 300 [330] 360 0 a 6 Use your calculator to find a sinI3rad b tan0.8rad © sinl2rad ds sing e cosy # wn In question 6, you don't need to convert each angle to degrees. You should set the angle mode on your calculator tor 7 CHALLENGE QUESTION e ‘Anna is told the size of angle BACin degrees and she is then asked to calculate en the length of the line segment BC. She uses her calculator but forgets that her calculator is in radian mode, Luckily, she still manages to obtain the correct answer. Given that angle BAC is between 10° and 15°, use graphing software to help you find the size of angle BAC in degrees correct to 2 decimal places. 8 Pees) You should already be familiar with the following mathematical words that are used in circle questions. Discuss and explain, with the aid of diagrams, the meaning of each of these words,8 Circular measure INTINUED Explain what is meant by: minor are and majar are * minor sector and major sector * minor segment and major segment. Ifyou know the radius, rem, and the angle 8 (in degrees) at the centre of the circle, describe how you would find: * arclenath * perimeter of sector * perimeter of segment + 212 of sector length of chord area of segment. 8.2 Length of an arc From the definition of a radian, the are that subtends an angle of I radian at the centre of a circle is of length r. Hence, if an are subtends an angle of @ radians at the centre, the length of the are is r ‘An are subtends an angle of & radians atthe centre of a circle with radius Sem, Find the length of the are in terms of 7. Answers, Are length =r =axt> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK A sector has an angle of 2 radians and an arc length of 9.6cm. Find the radius of the sector. Answers Are length = 70 96=1x2 r=48cm ‘The circle has radius Sem and centre O. o PQ is a tangent to the circle atthe point P. = RO isa straight ine. Find angle POQ, in radians b oR ¢ the perimeter of the shaded region, Answers. 2 tanrog=2 triangle OPO is right-angled since P@ is a tangent Angle POQ = tan”"(12) remember to have your calculator in radian mode = 1.176... = 1.18 radians b 0g*=12?+5* 00? = 169 ‘using Pythagoras’ theorem 00-13 Hence QR= Sem. © Perimeter=PQ+OR+arcPR —uscare PR=r0 = 1248+ (5% 1.176...) =25.9em Exercise 8.2 1 Find, in terms of a, the arc length of a circular sector of radius 6em and angle radius Sem and angle 2 € radius 10cm and angle 3 radius 18cm and angle = 180 >8 Circular measure Find the arc length of a circular sector of a radius 8m and angle 12 radians radius 2.5em and angle 0.8 radians Find, in radians, the angle of a circular sector of 2 radiuydom and arc length Sem radius 9em and are length 13.Sem. Find the perimeter of each of these circular sectors. a b € = = (As ABCD isa sectangle with Al jem and BC = 16cm. (isthe midpoint of BC: OAED is.a sector of a circle, centre O. Find a 40 bangle 40D, in radians ¢ the perimeter of the shaded region. Find a the length of arc AB b the fength of chord 4B © the perimeter of the shaded segment, ‘Triangle EPG is isosceles with g EG= FG=\6cm. GHis an ate of a circle, centre Fy tem 160m with angle HZFG = 0.85 radians. Find a the length of are GH a W F b thelength of EF 2 © the perimeter of the shaded region. >> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 8.3 Area of a sector To find the formula for the area of a sector you use the ratio: 8, area of sector angle in the sector area of circle ~ complete angle at the centre York When @ is measured in radians, the ratio becomes: A area of sector __ 9 ar Qn area of etor= 2x ar* Find the area of a sector ofa eiree with radius 6 cm and angle % radians Give your answer in terms of . 1 . 5% 8% 8x sin) = 29.0975. using area of triangle = 4absinc PRBB RD 64 using area of sector = 474 2 ‘Area of segment = area of sector AOB ~ area of triangle AOB = 64~ 29.0975. =§49cm?8 Circular measure Exercise 8.3 1. Find, in terms of 1, the area of a circular sector of 2 radius 6em and angle bb radius 15cm and angle 2 ¢ radius Oem and angle radius 9em and angle =. 2. Find the area of a circular sector of a radiusdem and angle 1.3 radians b radius 3.8em and angle 0.6 radians 3. Find, in radians, the angle of a circular sector of a radius 3em and area Sem” b radius Tem and area 30cm’. 4 POQisthe sector of a circle, centre O, radius 10cm. ‘The length of are PQ is Sem. Find a angle POQ, in radians b the area of the sector POO. 5 Asector of a circle, radius rem, has a perimeter of 150cm. Find an expression, in terms of , for the area of the sector. 6 ABCD isa rectangle with ‘AB=9em and BC= 18cm. is the midpoint of BC. OAED is asector of a circle, centre O. Find a AO bangle AOD, in radians © the area of the shaded region, 7 The circle has radius [2em and centre O. ‘PQisa tangent to the circle at the point P. QRO isa straight line. Find @ angle POQ, in radians bthearea of sector POR © the area of the shaded region.> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 8 AOBis the sector of a circle, centre O, radius 8em. 4 ACis a tangent to the circle at the point 4. (CBO is a straight line and the area of sector AOB is 32cm’, sem Find a angle AOB, in radians b thearea of triangle AOC é : € thearea of the shaded region. 9 Triangle EFGis isosceles with EG= FG =9em, g GHis an are of a circle, centre F, with angle HFG = 0.6 radians. Sem Sem Find 2 thearea of sector of HFG b theareaof tangle EFG © the area of the shaded region, 10 The diagram shows a circle, centre O, radius 12cm, nem. Angle AOB =? radians. Jia Are AB=9rem. 4 3n Ser Show that 0= 3 <> SN W F a b Find the area of the shaded region 11 AODisa sector of a circle, centre O, radius 4em. 6 BOC isa sector of a circle, centre 0, radius Lem. The shaded region has a perimeter of 18cm, Find a angle AOD, in radians b the area of the shaded region. 2 4 2 12, AOBis.a sector of a circle, centre O, with radius 9em. B Angle COD = 0.5 radians and angle ODC is a right angle, OC= Sem. Find LO as 0D Sem b © the perimeter of the A asad shaded region é 1 ! d_ the area of the shaded region. —$<— $4, —$ —__—*8 Circular measure 13 FOGisa seotor of a circle, centre O, with angle FOG=1.2 radians. = EOHTis a sector of circle, centre O, with radius Sem. ‘The shaded region has an area of 71.4em?, Find the perimeter of the shaded region. 44 CHALLENGE QUESTION ‘The diagram shows a semicircle, centre 0, radius 10em. FiLis the arc of a circle, centre E. Find the area of triangle ZOF b sector FOG © sector FEH the shaded region. 15 CHALLENGE QUESTION ‘The diagram shows a circle inscribed inside a square of side Jength 10cm. A quarter circle of radius 10cm is drawn with the vertex of the square as its centre, Find the shaded area. In this chapter you have leamed about using radians as @ unit of angular measure. 2 Without looking back in the textbook can you explain to a friend what a radian is? b What are the advantages of using radians as a unit of angular measure?> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK ‘One radian (1) is the size of the angle subtended at the centre of a circle, radius r, by an are of length r. ‘When @ is measured in radians: + the length of are 4B= 70 1 * thearea of sector AOB=51°# Past paper questions Worked example In this question all lengths are in centimetres, ‘The diagram shows a shaded shape. The arc 4 is the ‘major arc of a circle, centre , radius 10. The line AB is of length 15, the line OCis of length 25, and the lengths of AC and BC are equal. @ Show that the angle AOB is 1.70 radians correct to 2 decimal places. el b Find the perimeter of the shaded shape. ol ¢ Find the area of the shaded shape. 15) Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q10 Jun 2021 —EEEE]EE——_——LLL LEE 186 >8 Circular measure Answers : a sinsoc= 45 sind OC = 0,75 AOC= sin "0.95 AOB= 2% AOC=2* sin 0.75 = 1.6961... = 1.70102 dp Perimeter of shaded shape = are 4B + AC+ BC ate AB: are AB= 79 where Dis the angle in the major sector = 2x - AGB = 10 2m ~ 1.696..) = 45.8706... AC: AC? = 10? + 257-2 10% 25 * cosdC IC = 19.8565. BC: BC= AC= 19.8565... Perimeter of shavted shape = arc AB + AC + BC 45.8706... + 19.8565... + 19.8565... = 85.58...0m 85.6em to 3 sf € Aref shaded shape = 2 ® area of triangle OAC + area of major sector O4B winks igeiamabeal a 2x4 x 10x25 xsindOe+ br $x 10% 250.1544 x 10? x (25-408) 1 =2xdx 10% 25 0,75 +5 % 10? x (2a ~ 1.6961...) =2x = 187.5 + 229.35 Wem" to 3 sf 1 The diagram shows a sector OPG of a circle with centre O and radius xem, Angle PO is 0.8 radians. The point S lies on OD such that OS = Sem. The point R lies on OP such that angle ORS is a right angle Given that the area of triangle ORS is one-fifth of the area of sector OPQ, find> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 2 the area of sector OPO in terms of x and henee show that the value of x is 8.837, correct to 4 significant figures is] the perimeter of POSR B1 © the area of POSR. 2) Cambridge IGCSE Additional Mathematies 0606 Paper 21 QL Nov 2014 2 © A ae > ‘The diagram shows two circles, centres 4 and B, each of radius 10cm, The point # fies on the circumference of the circle with centre 4. The two circles intersect at the poims Cand D. The point £ lies on the circumference of the cizcle centre B such that ABE isa diameter. 1 Explain why triangle ABC is equitateral. 0 ii Write down, in terms of «, angle CBE. al iii Find the perimeter of the shaded region. (1 iv Find the area of the shaded region. ol Cambridge IGCSE Additional Mathematics 0606 Paper 11. O10 Nov 2015 3 P @ ‘The diagram shows a circle, centre O, radius Sem, The points P and Q lie on the circle. ‘The lines PT-and OF are tangents to the cirele and angle POQ = =F radians i Find the length of P7. a ji Find the area of the shaded region. 81 iii Find the perimeter of the shaded region. el Cambridge IGCSE Additional Mathematics 0606 Paper 21 OF Jun 20158 Circular measure <}. ‘The diagram shows a circle, centre O, radius rem. Points A, B and Care such that A and B lie on the circle and the tangents at A and B meet at C. Angle AOB = 0 radians, i Given that the area of the major sector AOB is 7 times the area of the minor sector AOB, find the value of 0. 2 fi Given also that the perimeter of the minor sector AOBis 20cm, show that the value of r, correct to 2 decimal places, is 7.18. el iii Using the values of @ and r from parts i and ii, find the perimeter of the shaded region ABC. BI iv Find the area of the shaded region ABC. Bi) Cambridge IGCSE Additional Mathematics 0606 Paper 12 Q9 Mar 2016 ‘The diagram shows a shape consisting of two circles of radius 3em and 4em with centres A and B which are Sem apart. The circles intersect at Cand D as shown. The lines AC and BC are tangents to the circles, centres B and A respectively, Find a the angle CAB in radians Qi b the perimeter of the whole shape 4) ¢ thearea of the whole shape. i] Cambridge IGCSE Additiona! Mathematics 0606 Paper 21 Q12 Nov 2020> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 6 AOBisasector ofa circle with centre O and radius 16em. 7 Angle 40B is radians. ‘The point Ces on OB such that OCis of length 7.Sem and AC is a straight lin, 16em a Find the perimeter of the shaded region. B b Find the area of the shaded region. 6 Cambridge IGCSE Additional Mathematics 0606 Paper 22 06 Mar 2021 7 a Acirele hasa radius 6em. A sector of this circle has a perimeter of 2(6 + Sa)em. Find the area of this sector. fa] By A Tem 0. 2 ‘The diagram shows the sector AOB of a circle with centre O and radius 7om. ‘Angle AOB = radians. Find the perimeter of the shaded region. BI) Cambridge IGCSE Additional Mathematics 0606 Paper 22 06 Mar 2020 8 The diagram shows a right-angled triangle ABC 5 with B= Remand angle ABC= F radians The points D and F lie on AC and BC respectively. BAD and ECD are sectors of the circles with centres 2a Sem Aand C respectively. Angle BA, radians, iz a i Find the area of the shaded regic iy ii Find the perimeter of the shaded region. BI Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q8 Jun 2019