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 Fig. 12.28). Find the area of the shaded region. (Use π = 3.14 and \( \sqrt{3}\) = 1.73205)

Asked by Pragya Singh | 1 year ago | 102

ABC is an equilateral triangle.

∠ A = ∠ B = ∠ C = 60°

There are three sectors each making 60°.

Area of ΔABC = 17320.5 cm^{2}

⇒ \(\dfrac{\sqrt{3}}{4}
\) ×(side)^{2} = 17320.5

⇒ (side)^{2} = \( 17320.5×\dfrac{4}{1.73205}
\)

⇒ (side)^{2} = 4×10^{4}

⇒ side = 200 cm

Radius of the circles = \( \dfrac{ 200}{2} \) cm = 100 cm

Area of the sector = (\(\dfrac{ 60°}{360°}
\))×πr^{2 }cm^{2}

= \(\dfrac{1}{6}\)×3.14×(100)^{2 }cm^{2}

= \(\dfrac{ 15700}{3}
\)cm^{2}

Area of 3 sectors = 3×\( \dfrac{ 15700}{3}
\) = 15700 cm^{2}

Thus, area of the shaded region

= Area of equilateral triangle ABC – Area of 3 sectors

= 17320.5-15700 cm^{2 }= 1620.5 cm^{2}

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