RevisionArc Length and Area of Sector

Question 1:

The diagram shows a quadrant of a circle, centre O and radius 8 cm. C is the midpoint of OB and a semicircle is drawn with OC as diameter. Find the perimeter of the shaded region, give your answer in the form ba .

Answers:

Arc AB 142 (8)4

Arc OC 122 (2)2

PerimeterOA+AB+BC+OC84 42126

RevisionArc Length and Area of Sector

Question 2:

In the diagram above, a regular pentagon ABCDE is inscribed in a circle.The center of the circle is O and OE = 14 cm. Calculatea) ∠AOBb) the length of the arc AE,c) the shaded area.

Answers:

a) 3 6 05

7 2

b) 7 23 6 0

2 1 4 5 35

1 7 . 6cm ( 3 s f )

c) Circle r 2 196

(3sf)14973.149

Pentagon-Circle=Shaded

)72sin141421(5Pentagon

2cm

RevisionArc Length and Area of Sector

Question 3:

The diagram below shows two circles, centre P and Q, of radius 8 cm and 2 cmrespectively, touching at point A. A common tangent touches the circles at B and at C.

Given that ∠��� = 50� ��� ∠��� = 130�

a) find the perimeter of the shaded region, leaving your answers in terms of .

b) find the area of the shaded region.

Answers:

a)

Per imeter = Arc AB + Arc AC + B C

2 29 1 4

9 8

3 23 8cm

b)

12 (8 2 ) (8) 40cm2

Sector BPA 8 89cm2

Sector AQC 1 49

Shaded 40 (8 89 1 4

9 ) 7.54cm2

RevisionArc Length and Area of Sector

Question 4:

The diagram shows a kite which is made of wire. The kite consists of a large semicirclewith diameter AB of 63 cm and two small semicircles of diameter 42 cm each. C and Dare the centres of the two small semicircles and

722

.

a) Explain why 60isEDC .b) Find

i. DE,ii. arc DFE,iii. the perimeter of the shaded region.

c) A special fabric is used to make the shaded region. Findi. the area of sector CED,ii. the area of segment DEF.

d) Show that the area enclosed by the fabric is 444 cm2, correct to 3 significant figures.

RevisionArc Length and Area of Sector

Answers:

a) C D 2 1cm; D E 2 1cmCE 21cmC&D are the ce ntre of the se micirclesC DE is eq uilateral

b)i. 2 1cm

ii. 6 03 6 0

2 2 1 2 2cm

iii. A R C A D + A r c C B + A r c A B 66669 9 231cm231 overlap o f 2 Arc DF E 187cm

c) 6 03 6 0

2 12 2 3 1cm 2

d) 2 3 1 C E D 4 0 . 0 4 1 3 9 8 40cm2 (3sf)

S emicirc le AB = 1559. 25cm2

S emicirc le AD = 693cm2

S emicirc le B C = 693cm2

Area of shade d figure = 1559.25 ( 693693 231) ( 40.0 ) 444 .29 444cm2 (3 sf)

RevisionArc Length and Area of Sector

Question 5:

The diagram shows a sector OAB of a circle with centre O where OA = 10 cm and theperimeter of the sector is 45 cm.

a) Show that radians5.2AOB .b) Hence find the shaded area.

Answers:

a) Arc AB45101025cm

r 25

2510

2.5rad

b) Calculator: Rad mode

12 r 2 1

2(10)(10)sin 2.5

121022.51

2(100)sin2.5

95.076 95.1cm2 (3sf)

RevisionArc Length and Area of Sector

Question 6:

In the diagram, O is the centre of the circle. P, Q, R, S are four points on the circumference of the circle and xPOR .

a) Find, in terms of x,i. PQR .ii. Find, in terms of x, PSR .

b) Given that obtuse radians1.2POR

and PO = 10 cm, calculatei. the length of major arc PQR.ii. the area of the minor segment PSR.

Answers:

a)

i. 12 x

ii. 1801

2 x

b)i. (2 2.1)1041.8cm (3sf)

ii. Calculator: Rad mode

12 r 2 1

2(10)(10)sin 2.5

Area of sector POR 1

2 102(2.1)

105cm2

POR 121010sin(2.1)

Area of segment 105POR61.839561.8cm2 (3sf)

RevisionArc Length and Area of Sector

Question 7:

The figure below shows a circle with centre O and radius of 7 cm.QRS and TUS are tangents to the circle at point R and U respectively.a) Show that rad84.1ROU .b) Calculate the perimeter of shaded segment.c) Calculate the area of shaded segment.

Answers:

a) 2 1.3 1.84159 1.84rad (3sf)

b) Calculator: Rad modeArc R Q 71. 84 159 12 .8 0113 (1)RU 2 7 2 7 2 2( 7) ( 7) cos R OU

RU 7 2 7 2 2( 7) ( 7) cos ROU (2)Perimeter (1) ( 2 ) 24. 036 24. 0cm

c) Calculator: Rad mode Sector ROUROU 1

2( 7) 2 (1 .8 4159 ) 1

2( 7) 2 sin 1. 84

21 .5cm2 ( 3sf )

RevisionArc Length and Area of Sector

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