Estimating Square Roots

Akhil has a square piece of cloth with an area of 125 $\text{cm}^2$. He wants to cut out the maximum size square handkerchief with an integer side length. First, we identify that 125 is not a perfect square. Next, we estimate its square root by finding the nearest perfect squares bounding 125. We know that the perfect squares closest to 125 are (which is $11^2$) and 144 (which is $12^2$). Since the area of the cloth is only 125 $\text{cm}^2$, we cannot cut out a square of area 144 $\text{cm}^2$. Therefore, the largest possible square handkerchief he can cut out must have an area of 121 $\text{cm}^2$. This means the maximum integer side length for the handkerchief is cm.