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Val123's avatar

When you find the area for a triangle, a rectangle, whatever, why is the final answer squared?

Asked by Val123 (12724points) May 13th, 2010

I know that it is, but I don’t understand why. What does squaring the answer tell us?

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24 Answers

gailcalled's avatar

@Val123: Are you sure about this? Area of a rectangle with sides a and b is axb. A exceptional rectangle is a square with the same length legs on all four sides. Area is a x a = a^2.

You have the radius squared when finding the area of a circle, but that is because of the definition of pi. (The number of times the radius can be laid around the circumference. So the area is the radius squared (the area of the square) x pi, which shaves off the areas between the square and circumscribed circle. (That was clear, I bet.)

Check out the site; the pictures make more sense.

http://www.mathsisfun.com/area.html

Val123's avatar

@gailcalled Actually, thanks for the pi explanation! That helped clear up a question I had earlier!

I’ve been thinking about it. To find the area of a rectangle you multiply the length times the width, lw=a. However, that just gives you half of the outside perimeter, one side and one top (or bottom). If you multiplied the answer, a, by 2, you’d get the entire outside perimeter. However, if you square the answer you get the whole outside perimeter and everything inside of it. The entire area….sound right?
Now, I have to go puzzle pi…..

Ivan's avatar

I think you mean the units. Let’s say we’re measuring length in meters, then any area will have the units ‘square-meters’, or ‘meters-squared’. It’s simply because area is a two-dimensional property, and a square meter is a unit of area; 1 square meter is 1 meter x 1 meter. If you treat the unit (meter) just as you would a number, you can see that your result is meters-squared (meters x meters). The same thing is true for volume, where the unit is meters-cubed (meters x meters x meters), because volume is a three-dimensional property.

hannahsugs's avatar

@Val123: your rectangle example isn’t correct. Half the perimeter would be L+W, not L*W.

As far as why the answer is “squared”, as @Ivan says, it’s the units that are squared. In a sense, you can think of “what’s the area of this rectangle/triangle/circle?” as actually asking: “how many little squares of length 1m by 1m (one square meter) will fit in this rectangle/triangle/circle?”

Therefore, you do the calculation and end up with an answer of, for example 15 “square meters”, or “15 squares with sides of length 1 meter would fit in this shape.”

hannahsugs's avatar

Alternately, you can think of units as following the same mathematical rules as numbers do. I think i can best illustrate this with some examples:

you can write:
10 meters + 5 meters
as
10*(1 meter) + 5*(1 meter)
which equals
(10+5)*(1 meter)
= 15 * (1 meter)
or 15 meters

likewise,
3 meters * 4 meters
can be written as
3*(1 meter) * 4 * (1 meter)
which equals
12 * (1 meter * 1 meter)
= 12 * (1 meter)^2
which we say as “12 meters squared”

Units also follow math rules outside of geometry. For example, if you travel 40 miles in 2 hours, your speed can be calculated by
(40 miles) / (2 hours)
= 40/2 (miles / hours)
= 20 miles/hour
which we say as “20 miles per hour”

SeventhSense's avatar

@Ivan
Yes like Ivan says it’s just the term to express area in two dimensional space- 5 square meters( 5 sq. m)

Val123's avatar

@Ivan Thanks Ivan! @hannahsugs Thank you too. Yes, when I thought about it more I realized that the perimeter would be L+H2.

Can someone draw me a picture?

hannahsugs's avatar

@Val123: a picture of what?

Val123's avatar

What you guys are telling me, about the units being squared. I know I’m being frustratingly literal, but if I have a rectangle 8 X 3, that’s telling me the area is 24^2. Which is 576. 576 what?

hannahsugs's avatar

No, the area of a rectangle with length 8×3 is 24. eight times three is 24.

I didn’t draw a picture, but I found one

That rectangle is 6×4, so the area is (coincidentally) also 24. There are 24 boxes enclosed in the rectangle. Now, about units: let’s assume that each of those boxes has a length of one inch. The area would then be 24 “inches squared”, meaning “there are 24 square inches (or boxes with sides length 1 inch) in the rectangle.”

Val123's avatar

That’s what my brain keeps telling me, that it’s simply 24, not 24^2. OK, what prompted this question: I recently started as a teacher with a HS Degree Completion program (not to be confused with a GED program), and I’m really re-learning everything I ever learned in HS and college as I go. I was showing a kid how to find the area of a given shape. However, the answer key suggested that whatever the answer was after multiplying L X H, that answer was squared. Now, the exponent was kind of ghosted, which only added to my confusion…..I asked the senior teacher what that meant, and she said, the area is always squared…..that just doesn’t make sense to me.

hannahsugs's avatar

It means that the unit is expressed as a “square,” just meaning that it’s 2-Dimensional area. I’m guessing that in the problem, there were no units, so the book probably put the exponent in there to remind you that it’s an area, not a length (no exponent) or a volume (which would be cubed), or something else.

The only other way I can think of saying this is that it’s “24” (pause) “inches squared,” not “24 inches” (pause) “squared” say it out loud, maybe it’ll make more sense?

Inches can be replaced by any unit, and in the case where the unit of measurement is not specified, as is the case with many problems in high school math, I would usually omit the “squared” part, or say “24 units squared.”

Val123's avatar

24 squares in the area. Right?

hannahsugs's avatar

yes, and those squares are 2-dimensional creatures, which is why we call them “units squared”.

Val123's avatar

That makes perfect sense. Well, what are your thoughts on the fact that they write the equation thusly:

L x H = X^2 (and the two is ghosted grey.)

I mean…why do they even DO that?

hannahsugs's avatar

While it’s not how I would write a textbook, i imagine they do it to remind you that the number has some physical meaning, and that an area of 24 is different from a length of 24. I agree that it’s potentially confusing.

Val123's avatar

It’s perfectly clear now! Thank you guys so much! However, one more question (because I need to be able to explain this to my kids students—have to quit calling them kids because, hail, one of my “kids” served in Vietnam.) Anyway, what happens when you get to figuring the area of, say a triangle, or an octagon. When dealing with a rectangle, it’s easy to imagine 24 little rectangles making the area. But…if you’re dealing with a triangle, are you imagining 24 little triangles creating the area? Because rectangles wouldn’t fit. Same question for octagons, etc. Tell Benny to stop sitting over the behind the curtain like the Great and Powerful Oz and LAUGHING at me!!

hannahsugs's avatar

No, it’s still little “squares.” If you drew a triangle on graph paper, like this one, you can see that there’s a certain number of whole squares enclosed, and some pieces of squares. Add up all the whole squares and square pieces, and you get the area.

Same thing with this octagon

Val123's avatar

I’m lookin’....but the squares (½ of a triangle) still break down into triangles at the edges in both examples….I’m pondering….the angle of the triangles are similar, I’m thinking…I’m looking…..I also been drinkin so I need to quit lookin’ and ponderin’ an thinkin’ for now! Although…I’ll have an epifany about 20 minutes from now, I just know it. Stay tuned! Tomorrow I’ll be interested to see what my Epiphany was!

gailcalled's avatar

@Val123: I’m getting confused myself, so if I am repeating someone else’s answer, sorry.

Use a right triangle as your paradigm model. Draw the entire square and then the line that would form the hypotenuse (from corner to opposite corner). Put in your grid system. If side A = 4 and side B = 5, you’d have 20 little squares; hence the area of your square is 20. Half of twenty is the area of your triangle (each of them). That is ½ x (A x B) = ½ (4×5) = ½ 20 = 10.

Forget the efiphany. Get some graph paper.

Tomorrow; Pythagoras, so keep the drawing.

SeventhSense's avatar

@Val123
You’re confusing yourself. Say you want to know the square footage of a rectangular room that is 8 feet in length x 4 feet in width by 8 feet in height . The area would represent the floor and that would be 32 square feet (8 length x4 width). That is a two dimensional space and would be the same for a flat piece of paper with the same dimensions.
The volume would be equivalent to 256 cubic feet(8 length x 4 width x 8 height). That is a 3 dimensional space. That would be physically represented as if you filled the room from floor to ceiling with 256 (1ft x 1 ft x 1ft boxes.)
The square or cubic measurement could be inches, meters or miles whichever unit of measurement you choose. Now obviously if it was a cube the volume could just be represented as a^(3 power) a = length of any side (or edge) since all sides are the same.

Val123's avatar

@hannahsugs AH!! Thank you! Thank you all. I’m really tempted to explain all this to my students, so they can understand the reasoning behind the ghosted ^2, and then tell them to disregard it…...
@SeventhSense Thank you guys!

SeventhSense's avatar

@Val123
It’s just me
Ok well there is the committee in my head

Val123's avatar

@SeventhSense LOL! I gotta committee in my head too! (Stop talking to me you guys!! Wait till I go to bed!!!)

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