The radius of a circle is measured to be 12.8 ± 0.2 cm. I need to find the Area and the Circumference of the circle? But how to I account for the uncertainties in each value?

Answer 1

The radius has a relative uncertainty of 0.2/12.8 = ± 1.6%

The circumference will have the same relative uncertainty: 1.6% .

The relative uncertainty in the area will be twice as much: 3.2%, because area depends upon the square of radius.

In differential terms:

A = pi r^2
ln A = ln pi + 2 ln r
dA/A = 2 dr/r

The formula is good for small relative errors (less than about 10%)