Significant figures when using formula

Question

I just want to clear something about significant figures.

Suppose we are asked something simple, to calculate the area of an octagon.

The formula is:

$$A=2(1+\sqrt{2})a^2$$

where $a$ is the side length.

Let's say $a=1.232$

Calculator gives $7.32870257$

Now the general rule for significant figures is that we should change the answer to use the least number of sig-figs in the original question.

There are two ways that I understand this:

• lowest # of sig-figs from equation, which in our case is just $1$, since 2 and 1 have one sig-fig

• lowest # of sig-figs from inputted data, since we entered the number $1.232$, and that was the only one we inputted, then the total number of sig-figs is actually $4$

Which is the correct one from above?

Using the first bullet would have a final answer of $7$, and using the second one would give $7.329$

Answer 1

The $2(1+\sqrt2)$ component should be treated as a constant with infinite number of significant digits (because there is no doubt about its value whatsoever). Since this constant is being multiplied with $a^2$, the answer should have the minimum number of significant digits among $2(1+\sqrt2 ),a$ and $a$. Hence the answer will have four significant digits.