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In the figure (ii) given below, ABC is an equilateral triangle of side 8 cm. A, B and C are the centers of circular arcs of equal radius. Find the area of the shaded region correct upto 2 decimal places.

In the figure, ABC is an equilateral triangle of side 8 cm. A, B and C are the centers of circular arcs of equal radius. Find the area of the shaded region correct upto 2 decimal places. Mensuration, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

Mensuration

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Answer

We know that

△ABC is an equilateral triangle of side 8 cm

A, B, C are the centres of three circular arcs of equal radius

Radius = 82\dfrac{8}{2} = 4 cm

By formula,

Area of △ABC = 34 side2\dfrac{\sqrt{3}}{4} \text{ side}^2

= 34\dfrac{\sqrt{3}}{4} × 8 × 8

= 34×64\dfrac{\sqrt{3}}{4} \times 64

= 16316\sqrt{3}

= 16 × 1.732

= 27.712 cm2.

So, the area of 3 equal sectors of 60° whose radius is 4 cm = 3 × πr2 × 60360\dfrac{60}{360}

= 3 × 3.142 × 4 × 4 × 16\dfrac{1}{6}

= 3.142 × 8

= 25.136 cm2.

Area of shaded region = Area of equilateral triangle - Area of 3 sectors

= 27.712 – 25.136

= 2.576 ≈ 2.58 cm2.

Hence, area of shaded region = 2.58 cm2.

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