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.
Best Answer
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 $1$.