Question

Find the area of a shaded region in the the following figure,where a circular arc of radius 7 cm has been drawn with vertex A of an equilateral triangle ABC of side 14 cm as centre.   (Use π = 22/7 and \[\sqrt{3}\] = 1.73)

 

Answer 1

In equilateral traingle all the angles are of  60°
∴ ∠BAC =  60°
 Area of the shaded region = (Area of triangle  ABC − Area of sector having central angle 60°) + Area of sector having central angle (360° − 60°)

\[= \frac{\sqrt{3}}{4} \left( AB \right)^2 - \frac{60°}{360°}\pi \left( 7 \right)^2 + \frac{300°}{360°}\pi \left( 7 \right)^2 \]
\[ = \frac{\sqrt{3}}{4} \left( 14 \right)^2 - \frac{1}{6} \times \frac{22}{7} \left( 7 \right)^2 + \frac{5}{6} \times \frac{22}{7} \left( 7 \right)^2 \]
\[ = 84 . 77 - 25 . 67 + 128 . 35\]
\[ = 187 . 45 {cm}^2\]

Hence, the area of shaded region is 187.45 cm²