How Do Significant Figures Affect Calculations in Physics Problems?

[zed101]

Hi! I'm confused, please I would love some help with this:

Homework Statement

if the inner radius of a ring is 3,56 cm and the outer radius is measured as 3,32 cm, compute the area of the ring

Homework Equations

When multiplying the number of significant figures stays the same, when adding or subtracting we keep same number of decimals.
Area of a circle: \pi r^2

The Attempt at a Solution

I'm basically subtracting $\pi r_1^{2}-\pi r_2^2$. When squaring the radii I get three significant figures and when subtrating I keep two decimals. The answer that way has two decimals for a total of three significant figures, my answer would be $\pi 1.65 cm^2$ (pi has infinite significant figures). However, if I rearrange the formula as $\pi (r_1+r_2)(r_1-r_2)$ for $r_1-r_2$ I get 0, 24 cm which only has two significant figures, this crops the total number of significant figures down to two when multiplied with $r_1+r_2$. What am I doing wrong?

 

[haruspex]

Errors can become quite large, in proportion, when taking differences of similarly sized numbers. Suppose each of the original values has an error of +/- 0.005. The range of possible values for the difference of their squares is 1.58 to 1.72, so it's not even two significant digits.
This affected your calculation when you took the difference of the squares. Squaring produced 4 digits for each value, of which one disappeared when you took the difference. But after squaring each you should, technically, only have kept 3 sig figures in each, so when the difference lost the high order digit you should only have had 2 sig figures left.

 

[I like Serena]

Welcome to PF, zed101!

You're not doing anything wrong.
The rule is only a rule of thumb.

In particular, it breaks down when subtracting quantities with a result close to zero.
This is what happens in your case.

 

[zed101]

Hi! thanks for the replies! :) I see, so the rule of thumb is not always true, teachers never said that. I suppose I should go with the first method before I hand in my paper. I'll add that note about the problem hehe.
Thanks! I think I can work safely now :)