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°)

$$\begin{gathered} =\frac{\sqrt{3}}{4}{\left(\mathrm{AB}\right)}^2-\frac{60°}{360°}\mathrm{\pi }{\left(7\right)}^2+\frac{300°}{360°}\mathrm{\pi }{\left(7\right)}^2 \\ =\frac{\sqrt{3}}{4}{\left(14\right)}^2-\frac{1}{6}\times \frac{22}{7}{\left(7\right)}^2+\frac{5}{6}\times \frac{22}{7}{\left(7\right)}^2 \\ =84.77-25.67+128.35 \\ =187.45{\mathrm{cm}}^2 \end{gathered}$$
=3√4(AB)2−60°360°π(6)2+300°360°π(6)2=1.734(12)2−16×3.14(6)2+56×3.14(6)2=62.28−18.84+94.2=137.64 cm2
Hence, the area of shaded region is 187.45 cm²