An error of 2 percent in excess is made while measuring the side of a square. What will be the percentage error in the calculated area of the square?

Introduction / Context:
This problem examines how a small percentage error in measuring a linear dimension affects the percentage error in an area calculation. The area of a square depends on the square of its side length, so a small error in side measurement can produce a larger error in the area. Understanding this relationship is important not only in exams but also in practical measurement tasks and engineering calculations.

Given Data / Assumptions:
- Let the true side length of the square be s units.
- There is an error of 2 percent in excess while measuring the side, so measured side is 1.02s.
- True area of the square is A true = s^2.
- Measured area of the square is A measured = (1.02s)^2.
- We need the percentage error in area, defined as (A measured - A true) / A true * 100 percent.

Concept / Approach:
When a measurement is changed by a small percentage, any quantity that depends on the square of that measurement changes approximately by twice that percentage, plus a small quadratic term. In this case, since area = side^2, we expect the area to increase by roughly 4 percent when the side increases by 2 percent. To be precise, we must square the factor 1.02 and then subtract 1 to find the exact relative increase in area and convert it into a percentage.

Step-by-Step Solution:
Step 1: Express the measured side as s measured = 1.02s. Step 2: True area is A true = s^2. Step 3: Measured area is A measured = (1.02s)^2 = 1.02^2 * s^2. Step 4: Compute 1.02^2 = 1.0404. Step 5: Therefore A measured = 1.0404 * s^2, meaning the area has increased by a factor of 1.0404. Step 6: The percentage error in area is (1.0404 - 1) * 100 percent = 0.0404 * 100 percent = 4.04 percent.

Verification / Alternative check:
As a rough mental check, recall that for small errors, the approximate error in area is about twice the error in side, so 2 percent in side suggests around 4 percent in area. The exact computation adds a small term equal to (2 percent)^2 = 0.04 percent, giving 4.04 percent, which matches our detailed calculation. This shows that the approximation rule is close but not exact, and the precise result is slightly higher than 4 percent.

Why Other Options Are Wrong:
2 percent is just the error in the side, not the area, and therefore underestimates the effect on area.
2.02 percent and 3.96 percent are incorrect approximations that do not correspond to squaring the factor 1.02.
4 percent ignores the additional small term from squaring 1.02 and is only an approximation, not the exact error asked in the question.

Common Pitfalls:
Many students assume the percentage error in area is simply double the percentage error in side and answer 4 percent without checking the exact calculation. Others square 2 instead of 1.02 and get completely wrong values. Always represent the new measurement as a multiplicative factor, compute the new derived quantity and then convert the relative change into a percentage.

Final Answer:
The percentage error in the calculated area of the square is 4.04 percent.