While measuring the side of a square, an error of 2% in excess is made (the measured side is 2% more than the true side). What is the percentage error in the calculated area of the square?

Introduction / Context: This question tests error propagation in measurement. For a square, area depends on the square of the side. When a quantity is squared, the relative (percentage) error approximately doubles for small errors. Here, because the side is 2% too large, the computed area becomes (1.02)^2 times the true area, leading to a precise percentage error of 4.04%.

Given Data / Assumptions:

• True side = s
• Measured side (2% excess) = 1.02s
• True area = s^2
• Calculated area = (1.02s)^2

Concept / Approach: Use scaling: calculated area / true area = (1.02s)^2 / s^2 = (1.02)^2. The percentage error is ((1.02)^2 - 1)*100%.

Step-by-Step Solution: Measured side = 1.02s Calculated area = (1.02s)^2 = (1.02)^2 * s^2 (1.02)^2 = 1.02*1.02 = 1.0404 So calculated area = 1.0404 * true area Percentage error = (1.0404 - 1)*100% = 0.0404*100% = 4.04%

Verification / Alternative check: Quick approximation rule: for area (square), % error ≈ 2*(% error in side) = 2*2% = 4%. The exact value 4.04% is close and confirms correctness with precise calculation.

Why Other Options Are Wrong: 1.04%: mistakenly treats 1.0404 as 1.0104 or forgets to convert properly to percent. 2.04% and 3.04%: come from partial multiplication or incorrect doubling rule usage. 4.00%: is only an approximation; the exact computed value is 4.04%.

Common Pitfalls: Assuming the area error is the same as side error. Using 1.02^2 incorrectly (e.g., writing 1.04 instead of 1.0404). Forgetting to multiply by 100 to express as percentage.

Final Answer: Percentage error in area = 4.04%