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The area of an equilateral triangle ABC is $$17320.5 cm^2$$. 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 figure). Find the area of the shaded region. (Use $$\pi \ = \ 3.14$$ and $$\sqrt{3} \ = \ 1.73205$$)

Let each side of the triangle be a cm. Then,

Area = 17320.5 cm2

$$\Rightarrow \ \frac{ \sqrt{3}}{4}a^2 \ = \ 17320.5$$
[$$\therefore$$ Area $$= \ \frac{ \sqrt{3}}{4}(side)^2$$]

$$\Rightarrow \ a^2 \ = \ \frac{17320.5 \ × \ 4}{ \sqrt{3}}$$ $$= \ 40000$$

$$\Rightarrow \ a \ = \ 200$$

$$\therefore$$ Radius of each circle = $$\frac{a}{2} = 100 cm.$$

Now, required area = Area of $$\triangle$$ ABC - 3 × (Area of a sector of angle 60º in a circle of 100 cm)

$$= \ 17320.5 \ - \ 3( \frac{60}{360} \ × \ 3.14 \ × \ 100 \ × \ 100)$$

$$= \ 17320.5 \ - \ 15700 \ = \ 1620.5$$ cm2