A circle is inscribed in a square of side 54 cms and another circle circumscribes the same square. Then the ratio of circumferences of the bigger circle to the smaller circle is

### Concept & Incircle and Circumcircle of a Square The radius of an inscribed circle (incircle) is half the side of the square. The radius of a circumscribed circle (circumcircle) is half the diagonal of the square. The ratio of circumferences is exactly the ratio of their radii. ### Step-by-Step Solution * Given: Side of the square $a = 54$ cm. * Radius of the smaller inscribed circle $r = \frac{a}{2} = 27$ cm. * Diagonal of the square is $a\sqrt{2} = 54\sqrt{2}$ cm. * Radius of the bigger circumscribed circle $R = \frac{a\sqrt{2}}{2} = 27\sqrt{2}$ cm. * The ratio of their circumferences is $\frac{2\pi R}{2\pi r} = \frac{R}{r}$. * Substituting the values: $\frac{27\sqrt{2}}{27} = \sqrt{2} : 1$. ### Exam Strategy & Shortcut For ANY square, the ratio of the circumcircle's radius to the incircle's radius is always $\sqrt{2} : 1$. The numerical side length (54 cm) is extra information designed to slow you down. The ratio of circumferences is directly $\sqrt{2} : 1$. ### Common Pitfall Students often calculate the numerical values of the circumferences before finding the ratio, which is time-consuming and introduces calculation errors. Cancel constants early! ### Final Answer Therefore, the correct answer is **$\sqrt{2} : 1$**.