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Question

The area of an equilateral triangle ABC is 17320.5 cm2. With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see Fig. 12.28). Find the area of the shaded region.
(Use π=3.14 and 3=1.73205)


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Solution

ABC is an equilateral triangle.
A=B=C=60
There are three sectors each making 60.

Area of equilateral Δ
=34×(side)2

Area of ΔABC=17320.5 cm2
34×(side)2=17320.5(side)2=17320.5×41.73205(side)2=4×104side=200 cm

Radius of the circles
=2002cm=100 cm

Area of the sector making angle θ
=(θ360)×πr2

Area of the sector
=(60360)×πr2 cm2
=16×3.14×(100)2 cm2
=157003 cm2

Area of 3 sectors
=3×157003=15700 cm2

Area of the shaded region = Area of equilateral triangle ABC - Area of 3 sectors
=17320.5 cm215700 cm2=1620.5 cm2

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