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I know there is an answer to a similar question here, however what I'm looking for is something slightly different.

So occasionally friends will come to me with mathematical problems that they need solving, and I try to anticipate the kinds of questions they might ask. Recently a friend needed to find the area of a bunch of rectangles so he knew how many tiles to buy for a wall he was tiling. One of the questions I thought he might ask was the title question, and I realised that, for all the time I've spent studying mathematics, I don't think I'd be able to give a decent intuitive explanation of this basic fact.

So my question is this: If you had to give a non-rigorous, intuitive explanation to a layman or young student, how would you do it?

Edit: A helpful comment and edit has suggested using the example of marbles as an explanation, however the reason why I don't think that fully answers my question is because I can anticipate that causing problems in someones intuition when confronted with a rectangle that has a decimal height or width. What would it mean to have $0.36$ marbles for instance?

Edit: To be clear, the question I'm asking here is how would you explain to a layman with little knowledge of maths why the area of a rectangle is the width times the height. I'm not asking how would I explain to my friend how many tiles he needs to fill his wall. I mentioned that problem simply because it's what motivated me to think of this question.

SeraPhim
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    Start with a horizontal line of length $l$. Now, drag the line upwards, keeping it horizontal. If you drag it up a distance of $b$, then you have ‘swept’ an area of $l\times b$. – Vishu Jul 27 '20 at 11:04
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    @TobyMak Somewhat, I did consider using the example of marbles etc, but I wonder if the conceptual jump from discrete objects to a rectangle which is more "continuous" might not translate well with someones intuition, especially for a layman with no mathematical knowledge whatsoever. I can anticipate someone being confused by the possibility of a rectangle of a height of say $0.36$, and then they might be confused by what $0.36$ marbles might mean. – SeraPhim Jul 27 '20 at 11:05
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    Related: "Area in axiomatic geometry" – Blue Jul 27 '20 at 11:06
  • I've found this question with better answers, but it doesn't sound like those answers are what you want. – Toby Mak Jul 27 '20 at 11:08
  • @TobyMak no not quite. I appreciate your help looking though! – SeraPhim Jul 27 '20 at 11:14
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    How many tiles your friend will need is actually a more complicated question than just measuring the area, because if the sides of the rectangle to be tiled are not integer multiples of the size of one tile, you have to break tiles to fit and you need to figure out how much of the area of the broken tiles you actually will be able to use. Tricks that will let you use fewer tiles, such as setting them at an oblique angle to the sides of the rectangle, may not be acceptable. – David K Jul 27 '20 at 11:30
  • $0.36$ of a square tile means you divide your tile into $10\times10=100$ smaller tiles and take 36 of these. – Intelligenti pauca Jul 27 '20 at 16:15
  • Does this answer your question? Area of rectangle. – Integrand Jul 28 '20 at 00:08
  • Any thoughts on the answers you've had, Sera? – Gerry Myerson Jul 29 '20 at 12:41
  • Several people have tried to help you here, Sera. Not polite to leave them hanging. – Gerry Myerson Jul 30 '20 at 13:04
  • @GerryMyerson I've been very busy the past few days. I've not had that much time to respond to everyone who's tried to help me. As you can see I've made numerous edits addressing a few recurring issues and responded to various peoples suggestions in comments. So I don't think it's fair to accuse me of leaving people hanging. I'll get to responding to others as soon as I'm able to devote some time to it. – SeraPhim Jul 30 '20 at 16:17
  • You have commented on one of the three answers, and your last comment and last edit were three or four days ago. I hope you can see how someone might conclude you were leaving people hanging. I also hope other pressures ease up on you, and you will have the time to get back to us. – Gerry Myerson Jul 30 '20 at 23:17
  • @GerryMyerson Well it doesn't really matter now cos I'm back so I'll respond to your question shortly. – SeraPhim Aug 02 '20 at 09:29
  • @Intelligentipauca The issue is here that I don't want to rely on the whole $1\times 1$ square has an area of $1$ thing. I'm trying to find an argument that doesn't really rely on that assertion. – SeraPhim Aug 02 '20 at 09:35
  • Also @Integrand I explain in my first edit why I'm not quite accepting that answer. It has a similar problem to the $1\times 1$ square has an area of $1$ assertion. – SeraPhim Aug 02 '20 at 09:35
  • Thanks for all your responses though guys, I appreciate your help! – SeraPhim Aug 02 '20 at 09:35

3 Answers3

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I think it really comes down to if you have a good idea of what area is or not. Area is a measure of how much space something takes up. 1 square unit is the area of a square of one unit by one unit. Now, a rectangle of lenght $a$ and width $b$ can be split into $a\times b$ squares of one unit by one unit.

Therefore, the area of a rectangle is $a\times b$.

A-Level Student
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  • This is assuming that the layperson or student in question has no grasp of what area really means. Not in a mathematical sense anyway. I'm not sure if the term "$1$ square unit is the area of a square of one unit by one unit" would be helpful for such a person. – SeraPhim Jul 27 '20 at 11:12
  • You don't think the layperson would understand it? – A-Level Student Jul 27 '20 at 11:16
  • The problem is that area is such a useful concept, not just as a practical concept, but as a concept that is the foundation for so many topics in mathematics, the sciences, and even extends to fields such as geography and economics (i.e the value of land). There's no point using the definitions of mathematical concepts, such as the Peano axioms, as those will be even more obscure. All disciplines are related and influenced by each other, so I'd say if they struggle with the concept of area, then they will have a hard time understanding anything else. – Toby Mak Jul 27 '20 at 11:17
  • @Toby mak, so do you think my answer is useful or too mathematical? – A-Level Student Jul 27 '20 at 11:19
  • @A-levelStudent well it depends on the layperson of course. I'm not suggesting lay people are stupid, don't misunderstand me. I just have a specific kind of person (based on my friend) in mind, who really has zero mathematical knowledge or intuition, but might be compelled to ask questions. – SeraPhim Jul 27 '20 at 11:20
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    @SeraPhim Call it a tile then instead of a 1x1 unit. The point is that if you make a room $a$ times as long, you need $a$ times as many tiles. The same if you make it $b$ times as wide. So a room that is $a$ times as long and $b$ times as wide as a single tile will need $a\times b$ tiles. – Jaap Scherphuis Jul 27 '20 at 11:21
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    I would urge them to take some introductory level maths courses, at something like an adult education centre. Not having a basic understanding of maths is like not having a basic understanding of English, which is not being able to write or read. Maths, like any other subject, is incredibly important for developing critical thinking skills and opens up so many opportunities for the learner. – Toby Mak Jul 27 '20 at 11:23
  • @A-levelStudent It's fine, but I feel you haven't fully addressed what 'area' means when you have fractional (rational number) units. – Toby Mak Jul 27 '20 at 11:24
  • @JaapScherphuis that has the same problem as the example with marbles that I spoke about in my edit. Splitting a rectangle into $1\times 1$ tile doesn't really make sense if you have a $0.54$ by $0.67$ rectangle. You can make it make sense of course by changing the scale of your tiles etc, but I think that starts getting into more advanced stuff that the kind of person I'm thinking of would find too abstract. – SeraPhim Jul 27 '20 at 11:25
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    @SeraPhim I don't think using a smaller tile size is too abstract. Bear in mind however that your question is in the context of a real situation where having to cut tiles down to make it fit does not automatically mean that the bits you cut off can be used elsewhere. In this practical situation you should probably simply round up each rectangle side to the next integer multiple of the tile-size anyway. – Jaap Scherphuis Jul 27 '20 at 11:30
  • @JaapScherphuis but what are you meant to say when you get asked the question "but why does a $1\times 1$ tile have an area of $1$?". You'd have to use the same reasoning at that point, that you can split that tile into smaller tiles, and so on ad infinitum. It doesn't really explain why the area is the height times the width. – SeraPhim Jul 27 '20 at 11:40
  • I'm playing devils advocate here by the way. I agree that in many cases your explanations would be acceptable. I'm just curious if there's a better way of explaining this, without resorting to axiomatically assigning a $1\times 1$ tile with an area of $1$. – SeraPhim Jul 27 '20 at 11:43
  • Because that is the definition of area. Area is a concept we have defined so that we can measure the real world (think about all the applications of integral calculus), and find relationships between different shapes. At some points, you have to accept the existence of certain fundamental concepts so that we can do more with the existing axioms of mathematics. If a child asks you 'why?' many, many times, chances are you will be unable to answer at some point, because what you need is quantitative reasoning, not qualitative reasoning. – Toby Mak Jul 27 '20 at 11:47
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    @SeraPhim The area of a tile doesn't really come into it. We're just counting how large the room is compared to a single tile so that we know how many tiles we need. That is why a tile is 1, because it is 1 tile. The actual area of a tile only becomes relevant if you want to compare prices of tiles of different sizes, and then you are probably still better off actually calculating the number of tiles needed in each case rather than just comparing by area. – Jaap Scherphuis Jul 27 '20 at 11:48
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    @TobyMak Yes I understand all those things, but a lay person might not. That's all I'm trying to get at. – SeraPhim Jul 27 '20 at 11:55
  • @JaapScherphuis the example of the tiles to fill a room was just for motivation for the more general concept of calculating area. This has nothing to do with how many tiles my friend needs to buy. – SeraPhim Jul 27 '20 at 11:56
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You certainly want the area to be proportional to the length, and also proportional to the height (since, e.g., a rectangle of twice the height can contain two copies of the smaller rectangle, so it must have twice the area). It follows that the area must be $cLH$, where $L$ is the length, $H$ is the height, and $c$ is a constant. Now, it really doesn't matter the slightest bit what (positive) value you take for $c$ (so long as you take the same value of $c$ for all rectangles), so we adopt the convention of taking it to be the simplest number around, which is $1$.

Gerry Myerson
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Here is how you would extend the concept of 'area' to the rational numbers:

Assuming the rectangle has dimensions $a \times b$ where $a,b$ are rational numbers, find the 'greatest common factor' of $a$ and $b$. For example, if the rectangle has width $0.80 = \frac{4}{5}$ and height $0.36 = \frac{9}{25}$, the lowest common multiple would be $\frac{\text{gcf}(4, 9)}{\text{gcf}(5, 25)} = \frac{1}{25} = 0.04$.

Then $0.80 = 0.04 \times 20$, and $0.36 = 0.04 \times 9$. Therefore, the area of this rectangle is $0.04 \times 0.04$ times that of a $20$ by $9$ rectangle, which you can subdivide into $20 \times 9$ unit squares, each of area $1$. Therefore, the area of this rectangle would be $0.04 \times 0.04 \times (20 \times 9) = 0.288$.

Toby Mak
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