Different Triangles With the Same Base and Height

Date: 04/16/2004 at 00:14:45
From: Maria
Subject: triangles with same base and height that have the same area

Why do two differently shaped triangles with the same base and height
have the same area?  I have been trying to figure this out by 
sketching it in different ways.  I can see it but I am having
difficulty explaining it.

Date: 04/16/2004 at 12:56:34
From: Doctor Peterson
Subject: Re: triangles with same base and height that have the same area

Hi, Maria.

I'm not sure what kind of answer you are looking for, since you say 
you "see it" (meaning that you believe by observation that it's 
true?), but can't "explain it" (meaning a proof, or just an intuitive 
sense of "why" or "how"?).  I'll show a couple ways to see how and why 
it works.

You know the formula for the area of a triangle,

  A = bh/2

which implies that the area depends only on the base and the height, 
so what you say HAS to be true, or the formula would be wrong.  Have 
you seen a proof of the formula?  Then that proves the claim you are 
questioning!

But you might like a more direct demonstration.

First, if you compare two triangles with the same base and height, 
like these,

        + - - - - - - - - - - - - - - - - - - - - - - -+    +
       /  \                                       /  /      |
      /     \                                /     /        |
     /        \                         /        /          |h
    /           \                  /           /            |
   /              \           /              /              |
  +-----------------+    +-----------------+                +
           b                      b

you can imagine sliding each particle of one horizontally to get it in 
the position of the other.  Each horizontal row is the same length in 
each triangle; so if you cut the left one into thin slices and shift 
them over, you will get something like the right one. The thinner the 
slices, the closer it will be:

        + - - - - - - - - - - - - - - - - - - - - - - -+    +
       +--+                                       +--+      |
      +-----+                                +-----+        |
     +--------+                         +--------+          |h
    +-----------+                  +-----------+            |
   +--------------+           +--------------+              |
  +-----------------+    +-----------------+                +
           b                      b

That approach is what calculus uses, with some more careful thinking 
to make it rigorously true.  See if you can work out some of the 
details.  This idea, taken into three dimensions, is called 
Cavalieri's Theorem.

This page shows the idea:

  Area of a Parallelogram
    http://mathforum.org/library/drmath/view/57790.html 

Another approach, a little more purely geometrical, is that you can 
actually cut one triangle apart into pieces and reassemble them to 
make the second triangle.  It's easier to see how you can put TWO of
the left hand triangle together to make a parallelogram, then
reassemble that into a rectangle, then turn that into a different
parallelogram that consists of two copies of the right hand triangle.
I'll show it by turning both triangles into the same rectangle.  We
add a second copy of each triangle:

        +-----------------+          +-----------------+
       /  \              /         /              /  /
      /     \           /        /           /     /
     /        \        /       /        /        /
    /           \     /      /     /           /
   /              \  /     /  /              /
  +-----------------+    +-----------------+

Now we cut the parallelograms vertically:

        +-----------------+          +-----------------+
       /|                /         / |               /
      / |               /        /   |             /
     /  |              /       /     |           /
    /   |             /      /       |         /
   /    |            /     /         |       /
  +-----+-----------+    +-----------+-----+

Now we move the left pieces to the right:

        +-----------------+          +-----------------+
        |                /|          |               / |
        |               / |          |             /   |
        |              /  |          |           /     |
        |             /   |          |         /       |
        |            /    |          |       /         |
        +-----------+-----+          +-----+-----------+

So their areas are clearly the same.

You may also be interested in Euclid's proof; see propositions 35-38 
on this page:

  Euclid's Elements, Book 1
    http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/17/2004 at 19:12:23
From: Maria
Subject: Thank you (triangles with same base and height that have the
same area)

Dr. Peterson,

Thank you very much!  Your visual explanation helped me to see how two
triangles with the same base and height must have the same area and be
able to explain it myself.  Now I understand WHY!