How is the percision you some measuring the length and width of a rectangle related to the precision of the resulting area measurement

Let 
a = height of the rectangle.
b = length of the rectangle.

Let x =  the fractional error per unit length.
That is the true measurement for length a can vary between (a-ax) and (a+ax).

Assume that measurement errors are maximum so that the measured values are a(1+x) and b(1+x).

Then the error in the area is
a(1+x)*b(1+x) - ab
= ab(1+x)² - ab
= ab(1 + 2x + x² - 1)
= ab(x² + 2x)

The fractional error in the area is
[ab(x² + 2x)/ab] = x² + 2x.

This means that the fractional error in the area is 
(a) doubled (or greater) when x<1,
(b) squared (or greater) when x≥1

If the fractional error in measuring the length is less than 1, then the error in the area will at least be doubled.

If the fractional error in measuring the length is 1 or greater, the error in the area will be squared. An error of more than 100% in measuring length is not likely.

The measurement error for the area will probably be doubled.