Given length $L = 12.4 \text{ cm}$. This measurement has 3 significant figures. Given width $W = 4.12 \text{ cm}$. This measurement also has 3 significant figures. The area is calculated by multiplying length and width: $A = L \times W = 12.4 \text{ cm} \times 4.12 \text{ cm} = 51.088 \text{ cm}^2$. When multiplying or dividing measurements, the final result should be rounded to the same number of significant figures as the measurement with the fewest significant figures. In this case, both measurements have 3 significant figures. Therefore, the area should be rounded to 3 significant figures. $51.088$ rounded to 3 significant figures is $51.1$ (since the first dropped digit, 8, is greater than or equal to 5, the preceding digit is rounded up).