Question

In the given figure, the area of the shaded portion APB is ?

• A. $\frac{1}{4}\pi r^2$
• B. $\frac{1}{4}(\pi -2)r^2$
• C. $\frac{1}{4}(\pi -1)r^2$
• D. None of these

Answer 1

The correct option is B $\frac{1}{4}(\pi -2)r^2$

Angle subtended by the arc = ${90}^{\circ }$
Area of the sector = $\frac{\theta }{{360}^{\circ }}\times \pi r^2$

Area of the sector = $\frac{90}{360}\pi r^2$

Area of the sector = $\frac{\pi r^2}{4}$ $c{m}^2$
Area of triangle = $\frac{1}{2}OA\times OB$

Area of triangle = $\frac{1}{2}r\times r$ = $\frac{r^2}{2}$
Thus, area of the segment = area of the sector - area of triangle

Area of segment = $\frac{\pi r^2}{4}-\frac{r^2}{2}$ = $\frac{1}{4}(\pi -2)r^2$.