Why is area measured in 'square' units?

[MrSmith]

My son asked me this the other day, and I had to pull out one of my "Dad answers".  Now I want a different perspective (possibly same answer).

 

Suppose we have a rectangle that measures 3 units wide by 4 units long.  Thus we know the area of the rectangle is 12 'square' units.

 

BUT...why does it have to be square units?  It could just as easily be 24 triangle units (if you cut the unit squiare in half)?  By the same token, couldn't any polygon which tessellates be used to measure area?  Why was the square chosen?

 

[Crimson Wife]

Because a square is the easiest polygon to calculate area for since it is just the side times itself. Triangles have a more complicated area formula.

[nmoira]

Probably because it's easiest to count squares and they, like regular triangles and hexagons fit together perfectly. Not all math programs use squares exclusively when counting the units of area: MEP also uses triangles... so maybe your son is on to something. :) Have him experiment with different shapes as the unit measurement. There are printables of triangle and hexagonal grid paper to be found.

[petepie2]

Area is a size measurement of a two-dimensional surface. Those dimensions (x and y) are perpendicular to each other, hence a square is formed. My two cents!

[nmoira]

Area is a size measurement of a two-dimensional surface. Those dimensions (x and y) are perpendicular to each other, hence a square is formed. My two cents!

 

This is true, but it's where we ended up rather than where we started. Using squares to conceptualize area predates Descartes.

 

ETA: corrected typo

[8filltheheart]

My son asked me this the other day, and I had to pull out one of my "Dad answers". Now I want a different perspective (possibly same answer).

 

Suppose we have a rectangle that measures 3 units wide by 4 units long. Thus we know the area of the rectangle is 12 'square' units.

 

BUT...why does it have to be square units? It could just as easily be 24 triangle units (if you cut the unit squiare in half)? By the same token, couldn't any polygon which tessellates be used to measure area? Why was the square chosen?

My answer is that it has nothing to do with shape??

 

Units times units equals units^2 and you refer to ^2 as squared which is why volume is referred to as cubic.

[cottonmama]

But it does have something to do with the shape.  If you're measuring area the simplest way, you'd make a grid and measure how many squares the object takes up.  The grid lines could be spaced out every inch ("square inches") or every centimeter (square centimeters), etc.

 

ETA: Okay, dh pointed out that at the most fundamental level, it's simply the multiplicative "squaring" of the units.  But that terminology probably evolved from the fact that the physical representation of multiplicative squaring *is* finding the area of a square.  It's kind of a chicken and egg question, but the point is, it's all related.

[8filltheheart]

But it does have something to do with the shape. If you're measuring area the simplest way, you'd make a grid and measure how many squares the object takes up. The grid lines could be spaced out every inch ("square inches") or every centimeter (square centimeters), etc.

True, which is why we probably refer to ^2 as squared. But the way the op phrased the question lead me to believe that she was missing the bigger pt of the units being multiplied.

 

Eta: You added your edit while I was posting. Yes, I agree with both you and your dh. :)

[cottonmama]

But the way the op phrased the question lead me to believe that she was missing the bigger pt of the units being multiplied.

 

Gotcha.  Actually, the more I think about it, the more I agree with you.  I would say to the op, even if we counted triangles to establish the area (which we could totally do), it wouldn't make sense to describe the result in terms of "triangle inches" -- we would say the figure is "24 triangles big" or something similar.  (And of course you'd have to establish the dimensions of the triangles you counted.)  Likewise, when we count squares, we could also say it's "12 squares big."  

 

But if we want to bring inches into it, we have to use square units because we're not even really talking about the squares or triangles we counted; we're talking about the multiplication we did of the units.  "Square inches" is, I guess, a lazy way of saying "inches squared."  You could also say "inch-inches."  But you can't say "inches traingled," right?  Because "triangling" isn't a mathematical operation like squaring is.

[Tanikit]

How old is your son? Squared in maths means something times itself - this is a bit of a chicken before the egg question since the term squared possibly does come from the fact that the area of a square is its length x breadth and they are the same.

5^2 = 5 x 5 = 25

so 1meter x 1 meter = 1m^2

It is best with a small child to show them the area of a square first - first one of side 1 and then one of side 2 (doesn't matter how big the measurement is as long as you double it for the one of side 2) but does he understand that 2 squared is 4? If he doesn't then use area to teach it - if he does then use squaring to show area. 

 

Triangle areas will not be standard since not all triangles are the same - they are not all equilateral or isosceles or whatever triangle depending on its angle, whereas a square of 1x1 is always just that. 

 

Draw it when you teach it - draw areas with squares then draw areas with various triangles and so on in them. 

 

I guess it is a philosophical question related to some arbitrary choice - like why we measure in inches or centimeters instead of paperclips or fingers (a cm is standard just like a square of 1unit will always have an area of 1unit^2 whereas a triangle with 2 sides of 1 unit does not always have an area of 1/2unit^2)

[cottonmama]

This is true, but it's where we ended up rather than where we started. Using squares to conceptualize area predates Descartes.

 

ETA: corrected typo

 

 

But I don't think the issue is the squares -- it's the units.  And I'm not sure the units of "square inches" existed before Descartes.  Euclid described area as a ratio -- this rectangle is so-many times the size of that square -- rather than in units with the multiplication built-in.  So if I'm not mistaken, the Cartesian coordinates were the entire reason we can say "inches squared" (which I now prefer over "square inches" because it's so much less likely to cause confusion between the multiplicative squaring and the grid squares).  Under Euclidean geometry, we could well swap out squares for triangles, because there are no units, just proportions.

 

 

 

This proposition is a generalization of the basic formula for the area of a rectangle, that is, the area of a rectangle is the product of its length and width. Such a formula depends on predetermined units of length and area so that the unit area is the area of a square whose sides have length equal to the unit length. Euclid and other Greek mathematicians did not use predetermined units of length or area, so they expressed this formula as a proportion. We would state that proportion as saying the ratio of the area of a given rectangle to the area of a given square is the product of the ratios of the lengths of the sides of the rectangle to the length of a side of the square. Of course, Euclid would say that without using the words 'area' and 'length' as follows: the ratio of the a given rectangle to a given square is the product of the ratios of the sides of the rectangle to a side of the square. Note that his terminology for a product of ratios involves "compounding the ratios." A natural generalization of the ratio of a rectangle to a square is the ratio of a rectangle to a rectangle. A broader generalization is the ratio of one parallelogram to another parallelogram having the same angles. That gives the generalization as stated in this proposition.

(Quote taken from http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI23.html.)

[nmoira]

But I don't think the issue is the squares -- it's the units.  And I'm not sure the units of "square inches" existed before Descartes.  Euclid described area as a ratio -- this rectangle is so-many times the size of that square -- rather than in units with the multiplication built-in.  So if I'm not mistaken, the Cartesian coordinates were the entire reason we can say "inches squared" (which I now prefer over "square inches" because it's so much less likely to cause confusion between the multiplicative squaring and the grid squares).  Under Euclidean geometry, we could well swap out squares for triangles, because there are no units, just proportions.

 

(Quote taken from http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI23.html.)

 

I'm thinking in more practical terms. Grids were laid out for art, architecture, and surveying. Property assigned, parceled, sold. Roads built. Etc. It's the same whether we use chains or poles or string: we've always used squares.

[CaffeineDiary]

The ancient Greeks (among other ancient civilizations) tended to solve math problems geometrically.  For example, if you have a square with sides of length 3 units, then the square will have an area of length times length ... which is to say, 3 times 3 = 9 square units.  Since this operation was performed so frequently, as a term of art whenever a number is multiplied by itself we refer to it as "squared".  For similar reasons, when a number is multiplied by itself, and then multiplied by itself again, we refer to it as being "cubed".

 

Referring to Geometry lessons, consider Pythagoras's Theorem, a^2 * b^2 = c^2.  We tend to think of this algebraically, but the easiest proof of it involves actually drawing squares on each side of a triangle and then rearranging them to demonstrate that the areas of the squares along sides a and b, added up, equal the size of the square along side c.

 

Vi Hart gives a visual demonstration of this proof, which should show viscerally why it's called "squared":

 

 

[kagmypts]

My answer is that it has nothing to do with shape??

 

Units times units equals units^2 and you refer to ^2 as squared which is why volume is referred to as cubic.

 

I agree with this.

[letsplaymath]

 

The ancient Greeks (among other ancient civilizations) tended to solve math problems geometrically.  For example, if you have a square with sides of length 3 units, then the square will have an area of length times length ... We tend to think of this algebraically ...

 

If you want to measure area, you need to use a flat shape. Think about what measuring means: comparing this thing we are interested in to a similar thing that we've agreed to use as a unit.

 

To measure length, you need to compare it to some other length, to see whether they mark off the same amount of distance. To measure volume, you compare it to some other volume, to see whether they hold the same amount of air/water/ether/whatever. To measure area, you compare your shape to some other flat shape to see whether they cover the same amount of flat space.

 

Now, what will be a convenient unit for counting off the amount of flat space something covers? Our unit should be:

• symmetrical, so it doesn't matter which way we lay it down
• space-filling, so we don't miss some part of the area we want to cover
• easily imagined or constructed, so we don't have to put too much effort into making a measurement

The first condition limits us to circles or regular polygons. The second condition eliminates the circle and limits us to polygons that can tessellate. Of these, the square is by far the easiest to work with.

 

These considerations do not depend on the system of measurement you are using (inches, cm, or cubits), but only on the nature of flat shapes. I don't know of any civilization that developed any other way of measuring flat space, and I doubt there ever was one. Even triangles, which seem like they should be a simpler shape than squares (having only three sides instead of four) are much more difficult to use as a measuring system, because it is harder to create fractions of triangles, and you know that measurements do not always come out in nice, even numbers.

 

Edited to add: I've read somewhere that Descartes did not originally insist that the axes of his coordinate geometry system be perpendicular. We are used to working with perpendicular axes, but everything would work just as well with a slant-wise system. (Try it and see!) In that case, I suppose we could measure area in rhombuses. I think it was the ease of working with squares that pushed history into using the perpendicular coordinate system we have today.

[MrSmith]

Hmm, I've never been referred to as a 'she' before.  I wonder if this means I can get the mom discount at Legoland?? :laugh:

 

Seriously though, this is why I like this forum - a simple question yields such an informative discussion :)  Thanks!

 

[mom2bee]

Because the unit itself is being multiplied with another, identical unit.

 

number/unit * number/unit = product of numbers/the unit raised to the 2nd power (or, in other words, squared).