A square is inscribed in a circle and another in a semi-circle of same radius. The ratio of the area of the first square to the area of the second square is

### Concept & Formula When a square is inscribed in a circle of radius $r$, its diagonal equals the diameter of the circle ($2r$). When a square is inscribed in a semicircle of radius $r$, the center of the semicircle lies at the midpoint of the base of the square. Area of a square given diagonal $d$ is: $$Area = \frac{d^2}{2}$$ ### Step-by-Step Solution Let the radius of the circle and semicircle be $r$. **1. Area of the first square (in a circle):** * The diagonal of the square = $2r$. * Area $A_1 = \frac{(2r)^2}{2} = \frac{4r^2}{2} = 2r^2$. **2. Area of the second square (in a semicircle):** * Let the side of this square be $x$. * The base lies on the diameter. The center of the semicircle is the midpoint of the base. * Forming a right triangle with the center, half the base ($\frac{x}{2}$), the side of the square ($x$), and the radius ($r$): $(\frac{x}{2})^2 + x^2 = r^2$ $\frac{x^2}{4} + x^2 = r^2$ $\frac{5x^2}{4} = r^2$ $x^2 = \frac{4r^2}{5}$ * The area of the second square $A_2 = x^2 = \frac{4r^2}{5}$. **3. Ratio of the areas:** * Ratio = $A_1 : A_2 = 2r^2 : \frac{4r^2}{5} = 2 : \frac{4}{5} = 10 : 4 = 5 : 2$. ### Exam Strategy & Shortcut If you memorize that a square inscribed in a semicircle has exactly $2/5$ the area of a square circumscribing the same circle ($4r^2$), and a square in a circle has half ($2r^2$), you can instantly set up the ratio $2r^2 / 0.8r^2 = 2.5 = 5:2$. ### Common Pitfall A frequent mistake is assuming the diagonal of the square inscribed in the semicircle is the radius $r$, which is incorrect. The radius is the hypotenuse of the triangle formed by half the base and the full height. ### Final Answer Therefore, the correct answer is **5 : 2**.