3.10 Percentage Error in Measurement

Percentage error compares a measured/estimated value against the true value, always taking the true value as the base:

%Error = (|Measured Value - True Value|)/(True Value) × 100

When errors occur in measuring the dimensions of a figure (length, breadth, radius), the resulting error in area or volume is found using the successive percentage change formula, since area/volume is a product of dimensions.

Worked Example 1: While measuring a rod, its length was recorded as 25.2 cm instead of the actual 25 cm. Find the percentage error. % Error = (0.2/25) × 100 = 0.8%.

Worked Example 2: While measuring the side of a square, a student makes an error of 5% in excess. Find the percentage error in the calculated area. Since Area = side × side, apply the successive formula with a = b = 5: % error in area = 5 + 5 + (5×5)/100 = 10 + 0.25 = 10.25% (in excess).

Worked Example 3: In measuring the sides of a rectangle, one side is taken 10% in excess and the other 5% in deficit. Find the percentage error in the calculated area. a = +10, b = −5. % error = 10 − 5 + (10 × −5)/100 = 5 − 0.5 = 4.5% in excess.

Worked Example 4: The radius of a circle is measured with an error of 2% in excess. Find the percentage error in the calculated area of the circle. Area = π × r², which depends on r twice (r × r), so treat this exactly like a square: a = b = 2. % error in area = 2 + 2 + (2×2)/100 = 4 + 0.04 = 4.04% in excess. This generalizes further: for a cube’s volume (side³), a length error of a% produces a volume error of approximately 3a% for small a (more precisely, apply the successive formula three times in a row).

Worked Example 5 (cuboid volume, three dimensions): The length, breadth, and height of a cuboid are each measured with an excess error of 5%. Find the percentage error in the calculated volume. Combine length and breadth first: a = b = 5 → 5 + 5 + (5×5)/100 = 10 + 0.25 = 10.25%. Now combine this 10.25% with the height’s 5% error: 10.25 + 5 + (10.25×5)/100 = 15.25 + 0.5125 = 15.7625% in excess.

Worked Example 6: While measuring a rectangular field, the length was measured 8% in excess and the breadth 4% in deficit. Find the percentage error in the calculated area, and state whether it is in excess or deficit. a = +8, b = −4. % error = 8 − 4 + (8 × −4)/100 = 4 − 0.32 = 3.68% in excess.