# What is the error propagation for $r = \sqrt{a^2+b^2}$ if the error in $\delta a$ and $\delta b$ are known?

Suppose I have some value $r$ which is defined as

$$r = \sqrt{a^2+b^2}$$

where $a$ and $b$ have errors of $\delta a$ and $\delta b$, respectively.

What is $\delta r$?

Using "standard" error propagation rules from this website, I proceed as follows:

$$\delta [a^2] = 2\frac{\delta a }{|a|}|a^2|$$ and similarly for $\delta [b^2]$.

Then we have

$$\delta [a^2+b^2] = \sqrt{[\delta[a^2]]^2+[\delta[b^2]]^2}$$ and $$\delta r = \delta[\sqrt{a^2+b^2}] = \frac{1}{2}\frac{\delta[a^2+b^2]}{|a^2+b^2|}|\sqrt{a^2+b^2}|$$

Substituting it all in we have:

$$\delta r = \frac{1}{2}\frac{\sqrt{\left [ 2\frac{\delta a}{|a|}|a^2|\right ]^2+\left [ 2\frac{\delta b}{|b|}|b^2|\right ]^2}}{|a^2+b^2|}\left | \sqrt{a^2+b^2}\right |$$

Is this correct?

It seems unbelievably complicated for the simple error of a sum of squares.

• You'd be surprised at how complicated error propagation can get. I haven't checked your work yet, but I wouldn't be surprised if that was correct. – Don Thousand Sep 7 '18 at 16:30