AREA OF A SHADED REGION WITH SECTOR OF A CIRCLE AND TRIANGLE

Find the area of the shaded region.

Problem 1 :

midpointgeometry1

The diagram shows an equilateral triangle ABC with sides of length 6 cm. 

P is the midpoint of AB.

Q is the midpoint of AC.

APQ is a sector of a circle, center A.

Calculate the area of the shaded region.

Give your answer correct to 3 significant figures.

Solution:

Area of shaded region=Area of⧍ABC-Area of sector(APQ)=12×a×b×sin θ-πr2θ360°=12×6×6×sin 60°-3.14×(3)2×60360=15.588-4.71=10.878 cm2

So, area of the shaded region is 10.878 cm2.

Problem 2 :

midpointgeometry2

The diagram shows a sector OABC of a circle with centre O.

OA = OC = 10.4 cm

Angle AOC = 120˚

(a) Calculate the length of the arc ABC of the sector. Give your answer correct to 3 significant figures.

(b) Calculate the area of the shaded segment ABC . Give your answer correct to 3 significant figures.

Solution:

a)

The length of arc ABC:

Arc length = 2πr (θ/360°)

= 2 × 3.14 × 10.4(120°/360°)

= 21.771 cm

b)

The area of shaded segment ABC,

Area of shaded region=πr2θ360°-12×a×b×sin θ=3.14×(10.4)2×120360-12×10.4×10.4×sin 120°=113.207-46.833=66.374cm2

Problem 3 :

The diagram shows a sector of a circle with centre O.

The radius of the circle is 8 cm.

PRS is an arc of the circle.

PS is a chord of the circle.

Angle POS = 40˚

midpointgeometry3

Calculate the area of the shaded segment.

Give your answer correct to 3 significant figures.

Solution:

Area of sector=θ360°×πr2Area of shaded region=Area of sector(POSR)-Area of⧍POS=πr2θ360°-12×a×b×sin θ=3.14×82×40360-12×8×8×sin 40°=22.328-20.576=1.752 cm2

Hence, area of shaded region is 1.752 cm2.

Problem 4 :

midpointgeometry4

ABC is an arc of a circle centre O with radius 80 m.

AC is a chord of the circle.

Angle AOC = 35˚.

Calculate the area of the shaded region.

Give your answer correct to 3 significant figures.

Solution:

Area of shaded region=Area of sector(OABCD)-Area of⧍OAC=πr2θ360°-12×a×b×sin θ=3.14×(80)2×35360-12×80×80×sin 35°=1955.556-1836.8=118.756 m2

Hence, area of shaded region is 118.756 m2.

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