What Trapezoid?

I disagree with the definition of trapezoid that was presented in Holt Geometry, the textbook my students are using. In it, trapezoid is defined as "a quadrilateral with exactly one pair of parallel sides." I believe that a trapezoid is "a quadrilateral with at least one pair of parallel sides."

Before I go any further, let me clarify that I am using the word trapezoid as Americans call this shape, which both the British and the Australian call trapezium.

If you Google definition of trapezoid, you will get the impression that mathematicians are starkly divided in their definitions of the term trapezoid. The truth is they are. One group leans toward the exclusive definition that Holt Geometry prefers, while the second group highly favors the inclusive definition. The second group is where I belong.

I found online a University of Washington  course syllabus for Math 444. While it adopted the inclusive definition, it explained that "the advantage of the (exclusive definition) is that it allows a verbal distinction between parallelograms and other quadrilaterals with some parallel sides." The same syllabus also explained that "the advantage of the inclusive definition is that any theorem proved for trapezoids is automatically a theorem for parallelograms." This simply means that if trapezoids have at least one pair of parallel sides, then every parallelogram easily qualifies as a trapezoid.

Proponents from both sides have their reasons. But the proponents for the inclusive definitions have gone as far as using higher level mathematics to explain themselves.

My own explanation as a proponent of inclusive definition is much more simple. Here it is:

Given Parallelograms 1, 2 and 3 as shown in the photo.

Parallelogram 1 is a 2" x 3" rectangle. Parallelogram 2 is a 3' x 3' square. Parallelogram 3 is a nonrectangle parallelogram with a base of 4 cm. and a height of 2 cm.

The universal formula that we use to determine the area of each of these parallelograms is A = bh. Applying this formula, we get areas of 6 square inches, 9 square feet, and 8 square centimeters, respectively.

Now, let us find the areas of the same figures but we will use instead the universal formula to determine the area of a trapezoid, A = (1/2)[b1 + b2]h.

For Parallelogram 1:
A = (1/2)[b1 + b2]h  = (1/2)[3" + 3"]2" = 6 square inches

For Parallelogram 2:
A = (1/2)[b1 + b2]h =  (1/2)[3' + 3']3' = 9 square feet

For Parallelogram 3:
A = (1/2)[b1 + b2]h = (1/2)[4 cm + 4 cm]2 cm = 8 square centimeters

The results are the same!

Clearly, the universal formula for the area of a trapezoid applies perfectly to each of the given parallelograms. This only shows that every parallelogram is a trapezoid and the inclusive definition is justified.

For practical purposes, I still go by our textbook's preferred definition though. But I make sure that my students are aware of the significance of the inclusive definition of trapezoid. This way, when they chance upon a college professor who is an ardent supporter of said definition, they would be okay.



If the inclusive definition of trapezoid is adopted, the resulting 2D classification chart would look like this.

To view an online discussion on this topic, visit The Math Forum's Ask Dr. Math.