If the measured side length of a square has a small error of 1%, what is the…

Concept

For a square with side length l, the area is A = l2.

For a small measurement error, the relative error of a power is approximately the exponent multiplied by the relative error of the measured quantity: dA/A ≈ 2(dl/l).

Application

1. Let the measured side length change from l to l(1 + 0.01).

2. Then the new area is l2(1 + 0.01)2.

3. Using the small-error rule, the approximate relative error in area is 2 × 1% = 2%.

Cross-check

An exact increase gives (1.01)2 − 1 = 0.0201 = 2.01%, while an exact decrease gives 1 − (0.99)2 = 1.99%. Both confirm that the first-order approximate error is 2%.

Therefore, the approximate percentage error in the area is 2%.