In regression analysis, if a potential third variable is a function of two others, is it (necessarily) redundant? Say I am using regression analysis to try and estimate the assembly time for widgets (which can be produced in various sizes) given some possible independent variables:


*

*Width

*Height

*Quantity of pieces (different sizes used as components)

*Did the American League win the World Series

*etc.


Most finished widgets are rectangular, however I'm wondering if the "squareness" of a widget explains some of the assembly time. That is, let's say Widget A is 2 feet wide and 18 feet long. Widget B is 6 feet square. Each have the same area (36 sq ft), but my hunch is that because of the assembly process, Widget B will take longer due to its square shape. 
Now to my question...
Since "squareness" is really just a calculated ratio of two other included variables (width and height), is it already "explained" in these two variables? Is it redundant to include "squareness" or similar ratio as an independent variable?
 A: I assume you are using ordinary least squares (linear) regression of the form
$$y_i=b_0+b_1 x_{1,i}+b_2 x_{2,i}+\ldots+\epsilon_i.$$
You are asking whether you have to include a predictor $\alpha$ if it is a function of another predictor $\beta$, i.e., if $\alpha=f(\beta)$.
The answer is: it depends on the functional form $f$. If $f$ is linear, i.e., $\alpha=c\beta$, then you should/can only include one (perfect collinearity)
$$y_i=b_0+b_1\alpha_i+b_2\beta_i+\epsilon_i=b_0+(b_2+b_1c)\beta+\epsilon_i.$$
But, as this shows, you can estimate the effect of $\alpha$ and $\beta$ both at once if you just include $\beta$, so this doesn't really hurt the quality of your fit.
However, since this is linear regression, you would lose information by including only one predictor if $f$ is nonlinear. Now, a ratio of two predictors is nonlinear, so you would lose information by not including the ratio as independent predictor.
So much on the theory. In practice, you should include "squareness" if you have good reasons to suspect it might affect your dependent variable (and I guess you do, otherwise you wouldn't ask), after controlling for height and width. Not including it in this case would be to introduce omitted variable bias deliberately. The only problem that might arise is that ratios can take very small or very large values, thus making standard errors very large (if you don't do significance testing, then maybe this is not a huge problem). But that could be solved by defining "squareness groups": any ratio below, say, 0.5 is group 1 (define dummy for that), any ratio $r\in(0.5,2]$ is group 2 (another dummy) and so on. 
