An error of 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is

### Concept & Error Propagation in Area When there is a small percentage error in measuring the linear dimension of a square, the error propagates in two dimensions. We can use the successive percentage change formula to find the net error in area: $$ \text{Net Error} = x + y + \frac{xy}{100} $$ Here, $x$ and $y$ represent the percentage error in length and width, which are both $+2\%$. ### Step-by-Step Solution * **Step 1:** The percentage error in measuring the side is $2\%$ in excess, meaning it is a $+2\%$ change. * **Step 2:** Let $x = 2$ and $y = 2$. * **Step 3:** Plug these values into the successive percentage formula: $\text{Error in Area} = 2 + 2 + \frac{2 \times 2}{100}$. * **Step 4:** Calculate the terms: $4 + \frac{4}{100} = 4 + 0.04$. * **Step 5:** The total percentage error is $4.04\%$. ### Exam Strategy & Shortcut Assume the true side is $100$. The true area is $100 \times 100 = 10,000$. With a $2\%$ excess error, the measured side is $102$. The calculated area is $102^2 = 10,404$. The excess area is $404$. Percentage error $= (\frac{404}{10000}) \times 100 = 4.04\%$. ### Common Pitfall A common mistake is simply adding the errors ($2\% + 2\% = 4\%$) or assuming the area error is identical to the side error ($2\%$). This ignores the compounding factor ($\frac{xy}{100}$). ### Final Answer Therefore, the correct answer is **4.04%**.