Question

Find the area of the shaded segment(in sq. units)

• A. $86.5$
• B. $89$
• C. $88.5$
• D. $90.5$

Answer 1

Answer: (C)

Area of shaded part = area sector OAPB - ar$\Delta OAB$

Area sector OAPB = $\frac{\theta }{{360}^0}\times \pi r^2$

= $\frac{{120}^0}{{360}^0}\times \frac{22}{7}\times 12\times 12$

$=150.86$ sq. units

Now, area $\Delta OAB$

Draw OM$\perp$AB, which results in two ${30}^0-{60}^0-{90}^0$ triangles, whose sides are in ratio $\frac{1}{2}:\frac{\sqrt{3}}{2}:1$

$\therefore$ In $\Delta OMA\Rightarrow OM=\frac{12}{2}=6,AM=\frac{12\sqrt{3}}{2}=6\sqrt{3},OA=12$

$\therefore$ area $\Delta OMA=\frac{1}{2}\times AM\times OM=\frac{1}{2}\times 6\sqrt{3}\times 6=18\sqrt{3}$

And area $\Delta OAB=2\times ar\Delta OMA=2\times 18\sqrt{3}=36\sqrt{3}=62.28$ sq. units

$\therefore$ Area of shaded region $=150.86-62.28=88.56$ sq.units