# Differentials - Area of a circle from its diameter

recently I've stumbled upon a differentails problem described below:

The area of a circle is calculated from the measurement of its diameter. Estimate the maximum percentual error in the measurement of the diameter if the percentual error on the area is 1%.

Usually we are asked the contrary - we have a radius or diameter error and we are asked for the error on the area/volume.

What I have (I'm not sure it is right) is

$$\frac{\Delta A}{A} \approx \frac{dA}{A} = \frac{\frac{\pi D \ dD}{2}}{\frac{\pi D^2}{4}} = 2D \ dD$$

Since $$dD = 1 \%$$ then $$\frac{dA}{A} = 0.5 \%$$

Am I on the right track?

Thank you.

Your first line should end with $$\frac{\frac{\pi D \ dD}{2}}{\frac{\pi D^2}{4}} = \frac{2 dD}{D}$$ so that overall you have that
$$\frac{dA}{A}=2\frac{dD}{D}$$ which gives you the answer you already had, i.e. that $$\frac{dD}{D}={1\over2}\frac{dA}{A}=0.5\%$$
Let's write $D$ for diameter. Then, if you double check your algebra, we have that
$$\frac{dA}{A} = 2 \frac{dD}{D}$$
Hence if $\frac{dA}{A} = 0.01$, it follows that $\frac{dD}{D} = 0.005$.