Using a Parallelogram to find the Area of a Triangle

Date: 06/26/2001 at 18:58:46
From: Christy V.
Subject: Math/area

What is the area of a triangle with a height of 12 inches and a base 
of 3 inches?

I tried it myself, but I just don't get it.

Date: 06/27/2001 at 17:12:03
From: Doctor Ian
Subject: Re: Math/area

Hi Christy,

Here's a way that you can figure this out when you've forgotten the
formula.  

First, draw the triangle. 

                . 
             . .   |
          .   .    | 12
       .     .     |
    .........      |
         3
         
Now draw a mirror image of the triangle, reflected across one side:

        .........
       .     . .   |
      .   .   .    | 12
     . .     .     |
    .........      |
         3


Now you have a parallelogram, so the area of the triangle must be 
exactly half the area of this parallelogram.

But what is the area of a parallelogram? Well, we can do a similar 
trick for any parallelogram:

    _____________
   |  /      .  /
   | /       . /   <-- cut this part off, and move it ___ 
   |/_________/                                          |
                                                         |
   ^                                                     |
   |_________________ here ______________________________|


So the area of any parallelogram is the same as the area of the 
rectangle with the same base and height. But the area of a rectangle 
is just the base times the height - so that means

  area(triangle) = (1/2) * area(parallelogram)

                 = (1/2) * area(rectangle)

                 = (1/2) * base * height

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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

Date: 06/29/2001 at 19:24:40
From: Christy V.
Subject: Re: math/area

Thank you, Doctor Math!