Question

In the given figure, a semicircle is drawn on the hypotenuse of the right-angled triangle. Another arc of radius 30 cm is drawn passing through A and C with O as the center. What is the area of the shaded region? (Take $\pi$ as 3.14 and $\sqrt{3}$ = 1.732)

• A. 117.75 sq. cm
• B. 216 sq. cm
• C. 271.95 sq. cm
• D. 108 sq. cm

Answer 1

The correct option is C

271.95 sq. cm

Area of the shaded region = Area of semicircle APC - Area of segment AQC
Area of semicircle = $\frac{\pi r^2}{2}=3.14\times 15\times \frac{15}{2}=353.25sq.cm$
Area of segment = Area of sector OAQC - Area of triangle OAC
Consider triangle OAC
Since all sides are equal to 30 cm in the triangle, it is an equilateral triangle.
Area of an equilateral triangle = $\frac{\sqrt{3}}{4}$ $\times$ $sid{e}^2$
=$\frac{\sqrt{3}}{4}$ $\times$ ${30}^2$
= 225$\sqrt{3}sq.cm=389.7sq.cm$
Area of the sector OAQC = $\frac{60}{360}\times \pi \times r^2=\frac{1}{6}\times 3.14\times 30\times 30=471sq.cm.$
Hence, area of segment AQC = 471 - 389.7 = 81.3 sq. cm
Hence, area of the shaded region = Area of semicircle - area of segment AQC
= 353.25 - 81.3 = 271.95 sq. cm.