Why is the area of a rectangle the height multiplied by the width?

Question

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.

Answer 1

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 $1$.

Answer 2

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 $a\times b$.

Answer 3

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 $0.04 \times 0.04 \times (20 \times 9) = 0.288$.