The area of the greatest circle which can be inscribed in a square whose perimeter is 120 cm, is

### Concept & Geometry When the greatest possible circle is inscribed inside a square, the diameter of the circle is exactly equal to the side length of the square. $$Diameter = Side\ of\ Square$$ $$Area = \pi r^2$$ ### Step-by-Step Solution * Given the perimeter of the square $= 120$ cm. * Calculate the side length of the square: $Side = \frac{Perimeter}{4} = \frac{120}{4} = 30$ cm. * The diameter of the inscribed circle equals the side of the square: $Diameter = 30$ cm. * Calculate the radius: $Radius (r) = \frac{30}{2} = 15$ cm. * Calculate the area of the circle: $Area = \pi r^2 = \frac{22}{7} \times (15)^2$ cm². ### Exam Strategy & Shortcut Always look at the options before calculating the final numerical answer. The options here are left in uncalculated format ($\pi \times r^2$). Once you determine the side is $30$, the radius is immediately $15$. You can instantly match this to option (d) without doing any multiplication. ### Common Pitfall Students often confuse the inscribed circle (diameter = side) with the circumscribed circle (diameter = diagonal of the square). Using the diagonal would result in an incorrectly larger radius. ### Final Answer Therefore, the correct answer is **$\frac{22}{7} \times (15)^2$ cm²**.