Why is the Area of a Triangle Always 1/2 Base Times Height?

[anantchowdhary]

How do we prove that the area of a triangle is \frac{1}{2} base*height for all bases of a particular triangle?
I mean to say...by definition of area..on a rectangle we can prove the area to be 1/2bh..but how do we prove that it will come out to be the same number for all ,base and altitude pairs of a particular triangle..

We can't use trigonometry(trig identities) as that comes from pythagora's theorem and pythagora's theorem comes from similarity of triangles...which in turn is derived using area of a triangle!

 

[Defennder]

The 1/2 comes from the fact that for any triangle you can always duplicate it to form a parallelogram.

 

[HallsofIvy]

How do we prove that the area of a triangle is \frac{1}{2} base*height for all bases of a particular triangle?
I mean to say...by definition of area..on a rectangle we can prove the area to be 1/2bh..but how do we prove that it will come out to be the same number for all ,base and altitude pairs of a particular triangle..

We can't use trigonometry(trig identities) as that comes from pythagora's theorem and pythagora's theorem comes from similarity of triangles...which in turn is derived using area of a triangle!

It's not clear what you are asking. The proof of the formula for area of a triangle does not assume any particular "base" and so any side could be used for the base. That is, the same proof applies to any "base" and "altitude".

In more detail, to prove that "area" of a triangle is (1/2)base*height, you first use the DEFINITION that the area of a rectangle is "width*height". You can use that to prove that the area of any parallelogram is "base*height" by dropping a perpendicular from one side vertex to an opposite side, then "moving" the triangle constructed to the other side, giving a rectangle.

Finally, given any triangle, choose any side to be a base and, on either of the other two sides, construct an identical triangle, but "inverted", show that you get a parallelogram. The area of the parallelogram is "base*height" and since it is made of two identical triangles, the area of each triangle is "(1/2)base*height".

Choose another side to be the base and do exactly the same thing: exactly the same proof shows that the area is "(1/2)base*height" for this base and height.

By the way: there are probably more proofs of the Pythagorean Theorem than any other single therorem: and many of them have nothing to do with "similar triangles" or "area". In addition, "similar" triangles are usually defined in terms of sides and angles, not area.

 

[anantchowdhary]

@ HallsOfIvy
Yes,I understand what you are saynig...But,my point is that how do we prove that in both cases the number 1/2*b*h are equal?

also,by using areas i meant...that we can prove the relation between ratio of sides in similar triangles using area of triangles..in particular the basic proportionality theorem.

Can you let me know of a proof of the pythagoras theorem that does not use areas...or similar triangles..in any way?

Thanks

 

[HallsofIvy]

All I can do is say what I said before: Doing exactly the same proof that (1/2)b*h gives the area, using each side as base in turn, proves that each of those numbers IS the area. Since the area of a triangle IS a single number, each of those must give that specific number!

http://www.jimloy.com/geometry/pythag.htm has several proofs of the Pythagorean theorem and refers to the book The Pythagorean Proposition, By Elisha Scott Loomis, which, he says, contains 256 different proofs!

The first proof on that website proves the Pythagorean theorem using similar triangles but NOT area.

I don't know what you mean by "that we can prove the relation between ratio of sides in similar triangles using area of triangles..in particular the basic proportionality theorem." or what you mean by the "basic proportionality theorem". Two plane figures, with the same number of sides and angles, are said to be "similar" if the angles are the same and the corresponding sides are in the same proportion. Perhaps what you mean by the "basic proportionality theorem" is that if a two triangles have the same angles, then they are similar. I do not believe that the proof of that requires areas.

 

[anantchowdhary]

Well...yes..that is what i meant by the basic proportionality theorem...I learned it by superposing one triangle over the other and then using areas...do you know of any other proof to derive the relation between sides(ratio)..?

Also,most proofs of the pythagora's theorem use area..
Now if we use the definition of area using the definite integral..i.e:cut up the figure into small strips of rectangles..and then add up all l*b of the small elements...using that how can we same without intuition hat the area's obtained by all 1/2*b*h are equal?

thanks

 

[anantchowdhary]

Someone please help!

 

[HallsofIvy]

Help with what? Your question has been answered repeatedly!

 

[anantchowdhary]

i was asking that how do you prove that the areas ''calculated'' will be the same...as area is just a definition...and if possible could you please tell me how we arrive upon relation of ratio of sides in similar triangles..without using areas?

 

[Werg22]

As far as I know, the additive property of area is an axiom independent of the uniqueness of area. Both are taken as axioms. When the area of a rectangle is defined to be its width times height, the area of a triangle is hb/2, using the additive property of area. By the uniqueness axiom, this number is the same whatever the choice of the base. So that the product of the height and the length of the base is constant in a triangle has to be taken as following from the derivation of the formula for the area rather than preceding it.

 

[anantchowdhary]

but what is the logic behind this axiom..it can't be on the basis of pure intuition...i quote wikipedia "It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts"

Now how do we mathematically prove this...this is what i am trying to ask!

 

[symbolipoint]

but what is the logic behind this axiom..it can't be on the basis of pure intuition...i quote wikipedia "It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts"

Now how do we mathematically prove this...this is what i am trying to ask!

No Proof! AXIOM. Yes Intuition is acceptable. Common critically found experience.

 

[anantchowdhary]

so this is all experimental??how do we say without proof that it WILL be valid in all cases?

 

[HallsofIvy]

No, it is not experimental. It not "intuition", at least not in the way I would use that word.

All of mathematics is based on "axioms". That is the very basis of mathematical proof. If, after all this, you can say "how do we say without proof that it WILL be valid in all cases?", I can only conclude that you have no concept of what a mathematical proof is.

You have been given repeated proofs that the formula for the area of a triangle is correct- and gives the same result no matter which side of the triangle is chosen as "base". I can only ask, what kind of answer do you WANT?

 

[anantchowdhary]

i repeatedly ask...where is the proof that both th forumulas give the same number??
you said that both of them WERE the area...but area is i had said a DEFINITION...so still it remains to prove that both numbers will be equal..
i quote wikipediaonce again.. "It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts"
also..if you could tell me about deriving a relation between sides of similar triangles..without using areas...as u had claimed earlier...

 

[Werg22]

In studying area, we assume we are given a function, called an area function, with domain the set of measurable surfaces and some special properties. On the basis of these properties (area axioms), we derive results about area, and results about the members of the set of measurable surfaces (e.g. that the product of the base and height in a triangle is a constant, regardless of the choice of the base). Constructing an area function is another matter. In studying the properties of area, nowhere do we assert that it's possible to define a function with such properties (know that it is though). You can investigate this on your own, https://www.amazon.com/dp/0201508672/?tag=pfamazon01-20 is a good place to look.

 

[HallsofIvy]

i repeatedly ask...where is the proof that both th forumulas give the same number??
you said that both of them WERE the area...but area is i had said a DEFINITION...so still it remains to prove that both numbers will be equal..
i quote wikipediaonce again.. "It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts"
also..if you could tell me about deriving a relation between sides of similar triangles..without using areas...as u had claimed earlier...

Perhaps it would help if you told us exactly what definition of area you are using?

The one I am using is this: "Area" is a set function that assigns to some plane figures a number such that
1) It is greater than or equal to 0.
2) If figure A is a subset of figure B, then the area of A is less than or equal to the area of B.
3) If figure A has area a, figure B has area b and their intersection contains only boundary points of A and B, the union of figures A and B has area a+ b.
4) If figure A is a square with sides of length 1, then the area of A is 1.

It can be shown that, for some sets, that implies a unique value. Those are the sets that we say "have area". It can be easily shown that all "geometric sets", that is squares, rectangles, parallelograms, circles, etc., have area. There are some sets, for example, the set of all irrational numbers between 0 and 1, that do NOT "have area".

 

[anantchowdhary]

I am simply using the definition:

Area of a rectangle is defined to be l*b.
Area of any other closed figure is defined as the limit evaluated by adding areas of n rectangles the figure is subdivided into where n-->infinity

 

[HallsofIvy]

Then it follows from that definition that if a set HAS an area, it is a unique number. From that it follows that it does not matter which side of a triangle you use as base, you will get the area: i.e. that unique number.

 

[anantchowdhary]

im sorry,i still don't completely understand...isnt there still a need to prove this?
as the set HAS an rea...according to evaluation of the limit using any PARTICULAR base in the definition known to me..

 

[symbolipoint]

anantchowdhary, are you trying to make this more complicated than it really is? We learn about proving the area formula for a triangle in college preparatory Geometry. That proof seems easily understandable. Definition of area; refer to square, define rectangle, area of rectangle; chop a triangle from one side and reposition on another part of figure to form parallogram; see the same height and same base; therefore same total area as before; cut parallelogram in half through two opposite corners; each has half the area of the original parallogram which was formed from the original rectangle.

The formula for area of triangle should cause no further problems. The concept is simple enough that a person does not really need college preparatory Geometry, but can clearly study the derivation as a Basic Mathematics student no higher than eighth grade.

 

[anantchowdhary]

@symbolipoint


well..it might seem very simple and very nice...but what i am saying is that area is DEFINED!Here we don't say anything about surfaces...and just by looking at the figure we CANNOT say that the area WILL be the same calculated by any means...Prove it mathematically to support your argument please...if you wish to do so...


Can you prove this,i quote wikipedia:"It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts".

?