General Area Formula

Date: 02/14/2002 at 22:46:39
From: Will Voorhees
Subject: General Area Formula

I heard that there was an all-inclusive formula for the area of a 
square, rectangle, parallelogram, trapezoid, and triangle.  Is this 
true?  I have looked everywhere, but I can't find anything.

Date: 02/15/2002 at 11:01:29
From: Doctor Ian
Subject: Re: General Area Formula

Hi Will,

Let's start with the most complicated (i.e., least symmetric) shape, 
which is a trapezoid:

       a
    ________
   /        \         area = height * (a+b)/2
  /__________\
       b


In the case where a and b are equal, we have a parallelogram:

       a
    ___________
   /          /       area = height * (a+b)/2
  /__________/
       b                   

And the formula still works.  Note that when a = b, 

  (a+b)/2 = (b+b)/2

          = 2*b/2

          = b

If we make all the angles square, we have a rectangle:

        a
    _________
   |         |        area = height * (a+b)/2
   |_________|
        b

And the formula still works.  If we make the width the same as the 
height, we have a square:

     a (= h)
    ___
   |   |              area = height * (a+b)/2
   |___|
     b

And the formula still works! The principal difference is that when you 
have a rectangle or a square, the height is trivial to find; while 
when you have a trapezoid or a parallelogram, the process can be 
somewhat more involved. 

So, what about a triangle?  Well, if we draw the triangle so that the 
base is horizontal, 

       a           a             
      /\          /|             a
     /  \        / |          . /
    /    \      /  |       .   /
   /______\    /___|     _____/
       b         b         b
    
then the value of the 'top base', a, is zero, so the formula gives us

  area = height * (a+b)/2

       = height * (0+b)/2

       = (1/2) height * b

So it works for a triangle, too - if you're willing to define a 
triangle as a quadrilateral with one zero-length side.

I hope this helps.  Thanks for an interesting question!

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

Date: 02/15/2002 at 22:46:24
From: Will Voorhees
Subject: General Area Formula

Thanks for the reply!  This really helped me a lot.

William Voorhees