1. **State the problem:** We have a square with an area of 225 cm², and we want to find the radius $r$ of the largest circle that can fit inside the square. 2. **Recall the formula for the area of a square:** $$\text{Area} = \text{side}^2$$ Since the area is 225 cm², we can find the side length $s$ of the square by: $$s = \sqrt{225}$$ 3. **Calculate the side length:** $$s = 15 \text{ cm}$$ 4. **Understand the relationship between the square and the inscribed circle:** The largest circle that fits inside the square touches all four sides, so its diameter equals the side length of the square. 5. **Express the diameter and radius of the circle:** $$\text{diameter} = s = 15 \text{ cm}$$ $$r = \frac{\text{diameter}}{2} = \frac{15}{2} = 7.5 \text{ cm}$$ 6. **Final answer:** The radius of the largest circle that can fit inside the square is $7.5$ cm.