How to find the error of the Pythagorean Theorem equation?

Question

I am trying to find the length of a hypotenuse with error, when the measurements of the two legs have an error.

So for this equation $$C = \sqrt{A^2+B^2}$$

when $A$ and $B$ each have an error of $\sigma_A$ and $\sigma_B$ respectively.

Answer 1

Using multivariable Taylor series we have $$\sqrt{(A\pm\sigma_A)^2+(B\pm\sigma_B)^2}=\sqrt{A^2+B^2}\pm\left(\sigma_A\frac{A}{\sqrt{A^2+B^2}}+\sigma_B\frac{B}{\sqrt{A^2+B^2}}\right)+O(\sigma_A^2+\sigma_B^2)$$ So one could say for small $\sigma_A$ and $\sigma_B$ that the error in $C$ is $$\sigma_C\approx\sigma_A\frac{A}{\sqrt{A^2+B^2}}+\sigma_B\frac{B}{\sqrt{A^2+B^2}}$$