Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Trapezoid: Visual Proof of Area Formula


Date: 04/11/98 at 09:29:50
From: Dwight
Subject: Trapezoid area formula proof

We were told that the area of a trapezoid is half the sum of the 
parallel sides x the height. How can I visually prove this formula?

I know that with a parallelogram you can cut off the triangle piece at 
one end and attach it at the other end and you have a rectangle again, 
so that proves why the formula area = base * height works. Is there 
a similar way to prove why the trapezoid formula works?


Date: 04/11/98 at 09:52:04
From: Doctor Jen
Subject: Re: Trapezoid area formula proof

Let's visualise the trapezoid as a rectangle and two triangles: 

            
         _____a______
        /|          |\      parallel sides lengths a and b
       / |          | \     height h
      /  |          |  \
     /___|__________|___\
              b


You can't cut off one triangle piece and attach it to the other (as 
you can with a parallelogram), because they may not be the same size. 

So we need to work out the areas of the separate bits. The area of the 
rectangle part is a * h.

The formula for the area of a triangle is (1/2) * base * height. 
If we have two triangles of the same height, we can say that the total 
area is (1/2) * firstbase * height + (1/2) * secondbase * height. 
This is the same as saying (1/2) * height * (firstbase + secondbase), 
because we can rearrange the formula for the total area. 

Well, in the trapezium, we don't know the lengths of the first base 
and the second base, but we do know that added together they make 
(b-a). We know this because b is the total length, and we've taken 
away a for the rectangle -- and what's left is the base lengths of 
both triangles. 

So the combined area of both triangles is 1/2 * (b-a) * h. 

The total area is the area of the triangles plus the rectangle, which 
is (1/2 * (b-a) * h) + a*h.

If you rearrange this you get the required formula 1/2 * (a+b) * h, 
and you've proven your result.

-Doctor Jen,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/