A circle is circumscribed around a square as shown in the figure. The area of one of the four shaded portions is equal to $\frac{4}{7}$. The radius of the circle is

### Concept & Areas of Segments The shaded regions are the four identical circular segments formed outside the square but inside the circle. The total shaded area is the area of the circle minus the area of the inscribed square. $$ \text{Total Shaded Area} = \pi r^2 - (\text{Side})^2 $$ ### Step-by-Step Solution * Let the radius of the circle be $r$. * The diagonal of the inscribed square is $2r$, so its side is $\frac{2r}{\sqrt{2}} = r\sqrt{2}$. * Area of the square $= (r\sqrt{2})^2 = 2r^2$. * Total area of all 4 shaded portions $= \pi r^2 - 2r^2 = r^2(\pi - 2)$. * Area of 1 shaded portion $= \frac{1}{4} r^2(\pi - 2)$. * We are given this area is $\frac{4}{7}$. So, $\frac{1}{4} r^2(\pi - 2) = \frac{4}{7}$. * Use $\pi \approx \frac{22}{7}$: $\pi - 2 = \frac{22}{7} - 2 = \frac{8}{7}$. * Equation becomes: $\frac{1}{4} r^2 \left(\frac{8}{7}\right) = \frac{4}{7} \implies \frac{2r^2}{7} = \frac{4}{7} \implies 2r^2 = 4 \implies r^2 = 2 \implies r = \sqrt{2}$. ### Exam Strategy & Shortcut Memorize that the total area between a circle and its inscribed square is $r^2(\pi - 2)$. One quarter of this is $\frac{r^2(\pi - 2)}{4}$. Substituting $\pi = 22/7$ makes the factor $2/7$, so $\frac{2}{7}r^2 = \frac{4}{7}$, immediately giving $r^2=2$. ### Common Pitfall A common pitfall is forgetting to divide the total area by 4, solving for the case where all four shaded regions equal $4/7$, which leads to an incorrect radius. ### Final Answer Therefore, the correct answer is **$\sqrt{2}$**.