# Calculating the uncertainty in an A4 page

## Homework Statement

Calculate the uncertainty in the area of an A4 page using a 30cm ruler with mm divisions.

## The Attempt at a Solution

Page is measured as:

Height H = 297mm

Width W = 210mm

H x W = 62370 mm^2

I'm using this formula to calculate the area uncertainty:

(dA/A)^2 = (dH/H)^2 + (dW/W)^2

(0.5/297)^2 + (0.5/210)^2 = (dA/62370)^2

(0.000002834 + 0.000005669)^0.5 = dA/62370

(0.000008503)^0.5 = dA/62370

0.002916 = dA/62370

dA = 181.87

So the area is:

62370 +/- 180 mm^2

Have I done this correctly?

Thank you.

You could have simplified your calculation if you observed that $(\frac {dH} {H})^2 + (\frac {dW} {W})^2 = (\frac {dA} {A})^2 = (\frac {dA} {HW})^2$, so you can multiply the equation by $(HW)^2$, getting $W^2dH^2 + H^2dW^2 = dA^2$. This has a very simple geometric interpretation, as $WdH$ and $HdW$ are the areas of rectangular margins at the sides of the paper during to uncertainty. Further noting that $dH = dW = dx = 0.5$, you get $dA = dx \sqrt {H^2 + W^2}$.