The uncertainty in a hypotenuse

You will very often have the lengths of two sides of a right angle, and need to compute the length of the hypotenuse. At other times, you'll have the components of a vector, and need to compute the magnitude of the vector. The calculation itself is easy:

But how can you compute the uncertainty in this result?

The answer is ... slowly and carefully. You can use the same old rules you've seen before

Let's do an example, to see how it works. Suppose we have

We'll use the notation ΔA to mean "the uncertainty in the quantity A". So we want to know

We can follow the simple rules, but it will take us three steps ...

Figure out the uncertainty in A² and B².

So we can figure out the uncertainty in the quantity A²

And end up with a value for A² with its uncertainty:

In exactly the same way, we can determine

Next, add together A² and B², with their uncertainties. This is easy: when adding two values, add their uncertainties.

Now, the fractional uncertainty in C² is around 4 percent:

Finally, we need to take the square root of C² in order to get our result, C. The rule is, if you raise a quantity to a power, you multiply its fractional uncertainty by that power. Here, we are raising C² to the 1/2 power, so we multiply its fractional uncertainty by 1/2:

So we can now compute the uncertainty in the hypotenuse:

At last! After all that work, we can now write the value of the hypotenuse together with its proper uncertainty.