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I cannot seem to comprehend why the dimensions of area are length squared. Area is the number of square units in a plane surface and is measured in terms of squares of sides of unit length. In short , I cannot fathom the meaning of a length times a length. Please shed some light on the matter .

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  • $\begingroup$ Squares have an area equal to length times length. I actually think that's one way they teach multiplication in K-12, at least in the U.S. $\endgroup$ – user228288 Jun 26 '15 at 0:11
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Here is a suggested walk down a path that might help make sense of the meaning of a "length times a length." There are two essential components at play here "length" and "area", which I'm sure you know.

First, to address "length." If you have two points in space, length is a measurement between the points. An important idea then comes into play - lines. While lines have many properties, one useful property is that if three points are in different places on the same line, then one of the three points will be "between" the others. This idea is important because it allows us to determine whether two points are "the same point" or different, and separated by some distance we call "length".

However, in a "plane" it is possible to have two points at the same place in one dimension, but not in another dimension. So if you create a line called "x" then two points can exist at the same place on "x" and yet not be at the same place in that plane or in space in general. On a line, they must be the same point. If they are not the same point then they are separated by some length by definition.

If you measure the distance between those points along a new line, then it is useful give that new line a new characteristic, which we call a "dimension." This idea is not basic at all, and in particular we call a line with all points on the same point of another line as "orthogonal" to the first line. Orthogonal is important to defining a second (or greater) dimension.

Now that we have developed the idea of more than one dimension, a question related to length comes up. For a line, when we ask "how far apart are these points?" we measure that with length. With 2 dimensional space, there is a similar question, "what do we call the space a shape occupies on a plane and how do we measure it?"

The answer is to go back to the concept of a line and define a unit of measure (a length that we call "1"). Then it is useful to create another definition, "if I have a line of unit 1 on the x dimension and another line of unit 1 on the y dimension, and join the endpoints using orthogonal lines of equal length, then let me define 'a unit of area' as something of this size in a plane."

I could draw a picture but I hope the words are sufficient.

EDIT:

However to expand on the concept of a unit of area, we can use notation for the x and y dimensions such that a point is denoted as "(x,y)" where x is the distance from some central location (zero) and the same is true of the y value (this is a standard notation). Using this the 4 points that define the unit of measure for a unit of area are (0,0), (0,1), (1,0) and (1,1).

In a single dimension, these points only have x or y values. Using the x dimension, these points are (0), (0), (1), (1). This means that using only a single dimension, the first two points are indistinguishable, along with the last two, and instead of four points there are only two. However, the points are not the same as we know because we must account for the second dimension. There is much to cover here, but we should talk about how it "looks" visually.

END EDIT.

Yes, it looks like a square of 1 unit. We need a way to talk about the "length" of objects in 2 dimensions. In 1 dimension, we have the unit length of a line segment. In 2 dimensions, we have a unit area of a planar shape.

However, the definition of a square comes from geometry, not algebra. So, it is easy to talk about area in terms of "square units" of some kind because of this likeness. Thus, we become lazy in general and talk about "squares" as being the units of area. And the idea of "multiplying lengths" is grammatical short-hand for "measure the 2 dimensional size for the shape where dimension x has X units and dimension y has Y units and use multiplication to determine the size of that area instead of adding up all of the units individually" Going back to the definition, it is easy to construct some of the images found as answers to this question.

Area as it relates to "multiplication" is kind of like thinking about the size of your home and "tape measures" or "rulers" - multiplication is the tool used to answer the question "how big" quickly and easily. To illustrate, you could take a piece of paper that is 1 unit by 1 unit and place it all over your house and add up the number of times you placed it in unique locations. Or, go back to the definition of "measuring size in 2 dimensional space as 1 unit of length along each dimension" and then realize that multiplication is a faster way to perform addition. Then you would realize you could measure orthogonal lengths in your house using a tape measure and use multiplication of those lengths to find your answer much more quickly...

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  • $\begingroup$ Thanks for your elaborate answer. Greatly appreciate it. But could you please elaborate how on a line x we can have two points that are on the same position on a line but not in a plane . :) $\endgroup$ – Musab Jun 26 '15 at 11:36
  • $\begingroup$ I edited my answer. I hope it helps. $\endgroup$ – Jim Jun 28 '15 at 0:52
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$$ \begin{array}{cc} \text{distance} = 3\text{ miles}\left\{ \vphantom{\begin{array}{c} \square \\ \square \\ \square \end{array}} \right. & \overbrace{ \begin{array}{|c|c|c|c|c|} \hline \square & \square & \square & \square & \square \\ \hline \square & \square & \square & \square & \square \\ \hline \square & \square & \square & \square & \square \\ \hline \end{array}}^{\displaystyle\text{distance} = 5\text{ miles}} \end{array} $$ $$5\text{ miles} \times 3\text{ miles} = 15\text{ square miles}.$$ You can count the $15$ squares, each of which is a square mile. You're multiplying a distance by a distance to get an area.

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In terms of "visually" seeing why using algebra, consider the following example...

To calculate the area of a triangle with $b = 3 \text{ in}$ and $h = 4 \text{ in}$, then the formula for area of a triangle is $$\begin{align}A_\triangle & = {1 \over 2}bh \\ & = {1 \over 2}(3 \text{ in})(4 \text{ in}) \\ & = {1 \over 2}(3)(4)\text{ in} \times \text{in} \\ & = 2(3)(\text{in})^2 \\ & = 6 \text{ in}^2\end{align}$$

You could apply this argument for any dimension of length.

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  • $\begingroup$ Does this mean that breadth and height are two different entities altogether ? $\endgroup$ – Musab Jun 26 '15 at 0:20
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easy question, just think of length 1 as a height and length 2 as a width so now you can comprehend the area as height times width

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Instead of thinking about length squared, think about squares with sides of unit length. For example, a square inch is a square with each side 1 inch long.

Imagine a rectangle that is 3 inches by 2 inches. You will be able to fit squares with each side 1 inch long in that rectangle. You will have 2 rows of 3 of these squares for a total of 6 squares with each side being an inch by an inch.

You can cover the rectangle with 6 squares inches (meaning 6 squares of this type with each side 1 inch long). 6 square inches is a measure of the area and is written 6 in$^2$, but is often read as 6 square inches.

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You can think of multiplying and dividing the units just as you do the numbers to determine the appropriate unit for your answer to the problem.

As you stated in your question the area of a plane surface is the number of "square-units" it contains. They are called "square-units" because the product was formed by multiplying not just the "pure numbers," but also the individual unit associated with each number. Thus applying your definition we get:

$(x–unit_x)\cdot(y–unit_y)$ $= (x\cdot y)–(unit_{x \cdot y})$

as you can now see the product is the result: $(x \cdot y)$ combined with the new unit: $(unit_{x\cdot y})$ which is equal to the square of the common unit of measurement used for both lengths and is also called the "square–units" that the plane surface contains; thus the number of those "square–units" is $x \cdot y $.

This is the meaning of length times length when the numbers represent a measurement of length and both measurements are expressed in the same unit i.e. square centimetres $(\text{cm}^2)$, square meters $(\text{m}^2)$, square inches $(\text{in.}^2)$, and square feet $(\text{ft}^2)$.

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