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Formula for the Area of a Trapezoid

Date: 01/14/2006 at 18:01:37
From: Keely
Subject: The area of a trapazoid.

How do you find the area of a trapezoid?  You have a page on area, but
you didn't discuss very well how to find the area of a trapezoid.  My
math teacher told us to picture it as a parallelogram; and this is the
formula he gave: (a+b)*1/2 h.

I frankly don't understand what he's talking about.  He says to 
imagine a hinge and that you are taking the top and flipping it over
and having it look like a parallogram:

    /                           \   The "o" represents the "hinge."
6" /_____________________________\o________________________
  /                               \                        /

He says you would take (20+25)*3.

Date: 01/14/2006 at 23:38:33
From: Doctor Peterson
Subject: Re: The area of a trapazoid.

Hi, Keely.

I'm not sure which page you looked at, or what you want to know.  It
sounds like you aren't really asking HOW (you have the formula) but
WHY that formula works, and in particular what in the world your 
teacher is talking about.  I wouldn't expect any old page on trapezoid
areas to answer that question!

We do give other explanations of the formula, such as

  Finding the Area of Squares and Triangles 

  Trapezoid: Visual Proof of Area Formula 

Here's a slightly different method related to your teacher's, which I
find very helpful:

We can put together TWO of the same trapezoid, one upside down, and
get a parallelogram:

             a           b
        /        \              /
      /           \           /
    /              \        /
           b           a

The area of the parallelogram is (a+b)h, where h is the height; so the
area of just one trapezoid is half that, or (a+b)h/2.

Your teacher's approach uses two halves of the trapezoid, instead of
half of two of them; it takes a little more geometrical knowledge to
appreciate it.  Cut my trapezoid in two along the line halfway up:

        /        \
    /              \

Now take the top and flip it over, as if it were attached by a hinge
at point o:

    /              \        /
            b          a

Notice a couple things about this. First, since the two angles at o
are supplementary, they will fit together just right when it's flipped
over, so that the top edges are now one straight line, and so are the
bottom edges.  (There's more to say to prove that, but I'm hoping you
can see that it at least makes sense.)  Similarly, the left and right
sides of the new figure are parallel.  So we have a parallelogram.

Now, the height of the parallelogram is half the height of the
original trapezoid, and the base is the sum of the top and bottom of
the trapezoid. So its area is

  A = (a+b)h/2

as before.

The picture also shows you, by the way, that the line you cut is half
of (a+b).  Do you see why?

It takes more to convince someone that this really works, but it makes
the formula itself very visual: you actually have a+b and h/2 as the
base and height.  So I can see why your teacher would like it.  It's
just a little hard to follow at first.     

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

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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