# Geometric Interpretation of Trapezoid Area Formula

###### Problem 1

Show that the area of a trapezoid (trapezium) is *half its height times the sum of the parallel sides*.

###### Solution

Cut the trapezoid along the midpoints of the pair of non-parallel side. This halves its height. Rotate one of the parts 180 degrees about a midpoint. In the figure below, it’s about the midpoint of DB. The resulting figure is a parallelogram. To get the area of the parallelogram, simply multiply its base and height.

###### Problem 2

Show that the area of a trapezoid (trapezium) PQRS is *half the sum of its parallel side times its height*.

###### Solution

The trick is to rotate PQRS 180 degrees about the midpoint of RS. This results to a parallelogram double the size of trapezoid PQRS. To get the area of the trapezoid, halve the area of the parallelogram.

Of course the two expressions for the area of a trapezoid are equivalent. A little algebra trick will produce identical expressions. What was shown here is that each expression has its own geometrical meaning or interpretation.

###### Problem 3

I have constructed two more lines parallel to side PQ as shown in the figure. What algebraic expressions for area of trapezoids can be constructed from the diagram below?

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I made a GeoGebra applet for Problems 1 and 2 and posted it in AgIMat.

Very nice! You might also like the pictorial overview on area in the Common Core State Standards posted on the Mathematics Teaching Community ( https://mathematicsteachingcommunity.math.uga.edu ) at https://mathematicsteachingcommunity.math.uga.edu/index.php/67/a-progression-on-area-in-the-ccss