The Geometric Intuition behind the Area Formula B*H=Area

[davidbenari]

I know not all shapes satisfy this relationship, but what is your intuition behind this formula that finds area (base*height=area). I think it would be best if we focus just on rectangles and squares since they seem to be the most elementary case.

I can think of a proof dealing with integer numbers, but it blows my mind how one can generalise this equation to include the irrational numbers. In other math forums they told it was just regarded as an axiom, but I'm pretty sure there must be a rigorous proof out there.

How do you prove this?

I was asked what my level was so I'll say it here: I've seen nothing beyond multivariable calculus.

Thanks a lot.

 

[olivermsun]

My intuition behind this is to divide up the square or rectangle into little squares and count them up.
If you have non integer sides you can divide up the leftovers into smaller squares. And so on. This procedure could be extended to irrational numbers.

You do this formally with limits in calculus.

 

[davidbenari]

I understand this, but I think something weird is going on. When we say 1m*1m =1m^2 and 2m*3m=6m^2, it seems there is something fundamental about sides and squares. I'm not sure if I'm being clear at all here, but I think there must be an intuition which is deeper than dividing a shape into little squares.

 

[Curious3141]

I know not all shapes satisfy this relationship, but what is your intuition behind this formula that finds area (base*height=area). I think it would be best if we focus just on rectangles and squares since they seem to be the most elementary case.

I can think of a proof dealing with integer numbers, but it blows my mind how one can generalise this equation to include the irrational numbers. In other math forums they told it was just regarded as an axiom, but I'm pretty sure there must be a rigorous proof out there.

How do you prove this?

I was asked what my level was so I'll say it here: I've seen nothing beyond multivariable calculus.

Thanks a lot.

You cross-posted this on Math SE, right? I was considering whether or not to add an answer or comment, but thought better of it, as the key question is the level of rigour you are seeking.

 

[olivermsun]

I understand this, but I think something weird is going on. When we say 1m*1m =1m^2 and 2m*3m=6m^2, it seems there is something fundamental about sides and squares. I'm not sure if I'm being clear at all here, but I think there must be an intuition which is deeper than dividing a shape into little squares.

IMHO it's a pretty deep result that 2 * 3 = 6 has a geometric interpretation like that.

What did you have in mind?

 

[davidbenari]

Curious3141: Yeah I did cross-post. I guess I'm not looking for extreme rigour since I don't know enough maths to do that; I guess I only want a deep-extremely-convincing-no-room-for-doubt interpretation of that equation.

olivermsun: I don't have anything really solid in my mind. I know what I'm going to say is really trivial but for some reason I think it points somewhere:
the unit square is by definition 1m^2, if have two sides 'a' and 'b' given in units 'm', there's nothing wrong with multiplying them. And hey! the answer has a geometric meaning! I don't know, haha, probably that sounds too stupid.

I do like the idea of dividing the shape up into unit squares, but I want to know about other interpretations.EDIT:

I guess another interpretation which is almost identical to dividing a shape up into unit squares is this one.

Suppose you have integer-valued sides 'A' and 'B'That implies this:

There exists at least A number of unit squares along the horizontal
There exists at least B number of unit squares along the vertical

Unit squares stack up perfectly, otherwise you will not form a rectangle.You start satisfying the first condition by setting up A squares along the horizontal. In order to form a rectangle, you must stack A squares on top of the first row until condition B is satisfied.

Multiplication is repeated addition. You added A number of squares B times = A*B.

This could maybe then be generalised for all positive numbers (I THINK, not sure). But I feel this doesn't address the fact that one dimensional stuff like 'm' is becoming two dimensional stuff like 'm^2', if that's clear at all.

Thanks for bearing with me.

 

[olivermsun]

Think of it another way.
You have x items. You have y items.
You have x*y possible combinations, which is kind of like the possible coordinates that make up the area x*y.

 

[WWGD]

Maybe it is the fact that n-cubes are product spaces, i.e.,$I^n =I^k \times I^{n-k}$ , for all m,n,k. This is not always true, e.g.,$S^n \neq S^k \times S^{n-k}$ for spheres.