In grade 10 we had a discussion with our mechanics and mathematics teacher about how should area be described. He insisted that area is a scalar, while some of us were totally convinced that it should be a vector. I, personally, was on the scalar side since vectors always have a direction, and area can not have such a characteristic. The only vector related to area is the normal to its surface, which is looked to me as a completely separate thing from area itself. It may be attached to it, but it is not what defines it.

However, now, my understanding of physics has changed a little. My current understanding of physics — and natural sciences in general — is that they are just models with the whole purpose of making it easier to understand, predict, and benefit from natural phenomena. Based on this definition of science itself, what we call vector and scalar are only abstract concepts we, humans, have created to facilitate analyzing physical systems; they do not relate whatsoever to the true nature of the object or quantity. Thus, questions about the relation of these abstractions with the observable real-life entities become less meaningful. These questions should instead be on how much does this specific abstraction benefit us in different cases.

So the question “Is area a vector or scalar?” becomes “Which could help us more? A vector or scalar area?”. This is way more approachable than the previous question and answering will be quite handy in actual problems instead of the philosophical perspective that we view nature with. It is not hard at all to answer this question, it just depends on the frame you are working in. That is, what I believe to be, the soul of science itself: to describe the unpredictable by our limited minds with the easiest possible approach, both computationally and intuitively.

If you deal with vector flows and fluxes that go through areas, and you calculate the final effect on area (or after passing through it), then it definitely helps to have a vector area; it makes many of the calculations you have to do a cross product of the quantity with a vector with the direction of the normal of the surface with a magnitude that of area. If we try to treat area as a scalar only in these cases it will be hell for us since we now have to write all equations with area and the sin of the angle between the normal of the surface and the vector quantity, both hard to write and read. This becomes crystal clear in electromagnetic systems. You always have some kind of flux being measured through some kind of area. You don’t have the time to get the angle of the surface with an axis, do you? Just cross product instead!

In contrast, there are countless simple systems where areas are no more than a straightforward multiplication with other quantities. It may not be the general formula for every case that is similar to this system, but it does the job for most day-to-day cases. There is no need to make area a vector when measuring the pressure of a gas on a cube. Like you would not prefer using a vector area when analyzing a fluid’s flow around a sphere. It would be a pain to think of the equations as a cross-product of vectors since they cancel into simple scalars anyway. What would you need angles and cross products for? Measuring the unmeasured angles?

To conclude, we can learn from this that context is the thing that really matters most of the time. Our mechanics teacher worked mostly on cases in which scalar area makes the most sense. Additionally, his strong pure mathematical background made him extend the — arguable — perfectionism and idealism of mathematics to the human-made physical models of the real world. This resulted in his certainty about his point. In his context, vectorized area means nothing, which is not a bad thing. We, as a society, should always recognize the difference in the correct points of view. It is possible, although hard, for us to accept the existence of different points of view. However, it is usually over one’s head to deeply realize the contextual correctness of other ideas that you may — currently — disagree with.