What is area?

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Two young mathematicians discuss the idea of area.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, I have a troubling question.
Riley: Yes?
Devyn: What is area?
Riley: Oh, for a rectangle it is just
\[ \text {height} \times \text {width} \]
Devyn: Right. But shouldn’t it mean more?

Area is a funny thing. What is it really? Well the idea is this:

Define something as having a “unit” area, and see how many times it “covers” something.

The most obivious thing to use for our “unit” area is a unit square. From this we can quickly move on to find the area of any rectangle as

\[ \text {height} \times \text {width} \]

Sometimes area is described as “how much paint will cover a surface.” Is this accurate? What do you think?

2025-01-06 19:18:48