Using Determinants to Find Area

Date: 06/09/98 at 20:52:31
From: Darren Goldner
Subject: Finding the area of triangles and parallelograms through the 
use of determinants

I'm a high school mathematics student at Stuyvesant in New York City. 
My math teacher has assigned me the task of reporting upon how the 
area of a triangle and that of a parallelogram can be found through 
the use of determinants. I have no idea where to start. Can you give 
me the formulae and explanations? 

Thank you for your attention.

Date: 06/09/98 at 23:35:51
From: Doctor Pat
Subject: Re: Finding the area of triangles and parallelograms through 
the use of determinants

Darren,

If the triangle or parallelogram is in the coordinate plane you can 
find the area using just the three vertices of the triangle (or three 
adjacent vertices of the parallelogram) in a 3x3 matrix.   

For the points (xa,ya), (xb,yb), (xc,yc), you can just put the x and y 
values in as the first two rows of the matrix, then fill the third row 
with ones:  

   |  xa    xb    xc  |
   |  ya    yb    yc  |
   |   1     1     1  |

The absolute value of the determinant is the area of the 
parallelogram, and 1/2 that value is the area of the triangle.   
    
There is also a way to do this with vectors that involves what is 
called the cross product. If you have the desire to pursue more ways 
to do this, why not use the search engine here at the Math Forum to 
search for key words like "vector cross product"? The searcher is at

  http://mathforum.org/grepform.html    

There are lots of interesting ideas in math that a knowledge of 
vectors can help you to understand. 

Good luck,
                  
-Doctor Pat,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/