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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: 

        /|          |\      parallel sides lengths a and b
       / |          | \     height h
      /  |          |  \

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
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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