How 'x' (multiplication) found its way into formula of area of rectangle?

[pairofstrings]

Hi, I know that area of rectangle is length x breadth. I tried to find proof of area of rectangle but I found that the proof was solved by taking formula of area of square into consideration. But what I don't understand is why area of rectangle should be length times breadth, or side times side as in case of area of square.
Question is:
1. How do I know that area of rectangle should be:
length x breadth (what makes 'x' qualify for the formula of area of rectangle or square?)
and not:
length + breadth or anything else.
2. What is 'x' really?
Thanks.

 

[jbriggs444]

How would you define the "area" of an arbitrary shape if you were making the rules?

 

[HallsofIvy]

To answer your last question first, "what is 'x', really"- it is a symbol meaning multiplication! Surely you knew that so I'm not sure what your question really is. (Since "x" is quite commonly used as a "variable" in algebra and beyond it is typically not used to indicate multiplication in algebra or beyond- use * or juxtaposition to indicate multiplication: h*w or just hw.)

As to why the area of a rectangle is "length times height" it is for two reasons:
1) Area is defined that way! An area of "one square meter" is defined as the area of a square one meter on a side.

2) Multiplication is defined that way! If you have 3 boxes, each containing 4 objects, then you have 4+ 4+ 4= 12 objects and we define multiplication to be repeated addition: 3 times 4= 12.

Now, put those together. If you have a rectangle that is 6 meters in length (horizontally) and 4 meteres in height (vertically), measuring each meter horizontally and drawing a vertical line at that point, you divide the 6 meter length into 6 section, each one meter long. If you divide the vertical height in the same way, one horizontal line at each meter, you divide that 4 meter height into 4 sections, each one meter long. But you have divided the area into four rows, each row consisting of 6 one meter by meter inch squares. That is, we have four "boxes" (the four rows) each containing 6 objects, 6 one meter by one meter squares. By the "definition of multiplication" in (2) above, there are 4(6)= 24 such boxes. By the "definition of one square meter" in (1) above. each of those squares has area 1 square meter. 24 of them will have area 24 square meters.

 

[pairofstrings]

In my recent attempt to find area of rectangle, I drew a figure of rectangle 6 units wide and 3 units tall.

In my attempt I assumed that I am traveling by foot. Each unit is one step. I travel 3 steps vertically and 1 step horizontally while I cover the area. I took 27 steps to move from initial point to final point. If I reached the end point this way it means that I have covered the entire area. True?
I believe, the following is a false statement.
27 steps = 6 x 3 => 27 steps = 18 units (false). 6 x 3 represents the formula of area of rectangle by proven method.
I later decided to convert my 27 steps into something which fits my equation.
So, 27 steps = z = 18 units. What could be z? My thought is that if z could give me the exact value of area of given rectangle (which is 18 units) then I somehow need to reduce 27 by some amount to make it 18 units. What could be that amount and how do I know how to calculate?
Does all of this make sense or my logic is flawed?
If flawed then how to calculate area by traveling by foot!? If traveling by foot logic to find area of rectangle is flawed, please explain why.

 

[PeroK]

Hi, I know that area of rectangle is length x breadth. I tried to find proof of area of rectangle but I found that the proof was solved by taking formula of area of square into consideration. But what I don't understand is why area of rectangle should be length times breadth, or side times side as in case of area of square.
Question is:
1. How do I know that area of rectangle should be:
length x breadth (what makes 'x' qualify for the formula of area of rectangle or square?)
and not:
length + breadth or anything else.
2. What is 'x' really?
Thanks.

I think most people would see that the area of an square or rectangle is l x b by definition. But, let's assume you don't see this. There is another way to calculate an area, which is to pick random points in a larger area and see how many are in the smaller area.

This will give you a relative measure of two areas. It will show you that a rectangle twice the length will have twice the area; a rectangle with twice the width will have twice the area; and a rectangle both twice as long and twice as wide will have 4 times the area.

In your diagram it would confirm that the area is 18 times larger than a single 1x1 unit.

This might lead you the conclusion that area might be well-defined as length x breadth.

 

[Mark44]

In my recent attempt to find area of rectangle, I drew a figure of rectangle 6 units wide and 3 units tall.
View attachment 74883
In my attempt I assumed that I am traveling by foot. Each unit is one step. I travel 3 steps vertically and 1 step horizontally while I cover the area.

You are not "covering the area". You are following what is essentially a one-dimensional path (length) through a two-dimensional region (area).

I took 27 steps to move from initial point to final point. If I reached the end point this way it means that I have covered the entire area. True?

No, false. Clearly the lines you drew do not cover each point in each of the rectangles. You are trying to measure something that is two dimensional using a one-dimensional measure. There is an essential difference between one-D measures such as meter (m) and two-D measures such as square meters (m²).

I believe, the following is a false statement.
27 steps = 6 x 3 => 27 steps = 18 units (false).

Yes, this is false. You are completely ignoring the units. Your steps are linear (one-D), and the area is in units of some two-D measure such as square inches or square meters.

6 x 3 represents the formula of area of rectangle by proven method.
I later decided to convert my 27 steps into something which fits my equation.
So, 27 steps = z = 18 units.

As mentioned above, you're comparing apples and oranges. As already mentioned, you are ignoring the units of area.

What could be z? My thought is that if z could give me the exact value of area of given rectangle (which is 18 units) then I somehow need to reduce 27 by some amount to make it 18 units. What could be that amount and how do I know how to calculate?
Does all of this make sense or my logic is flawed?
If flawed then how to calculate area by traveling by foot!? If traveling by foot logic to find area of rectangle is flawed, please explain why.

When you measure something, you need to use a measuring device that is appropriate to the thing you're measuring. You wouldn't use a ruler to measure the weight of a rock, nor would you use a stopwatch to determine the color of an apple.

If you want to measure a length, you can use a ruler. If you want to measure an area, you need to use something that is two-dimensional. If you want to measure the volume of, say, a box, you would use some three-dimensional object to see how many of them would fit into the box.

 

[jbriggs444]

Note that the "27 step" path visits each corner in a grid of 3 squares by 6 exactly once. The same result could be computed more easily by multiplying 4 by 7 and subtracting one.

The term "fencepost error" may be of interest.