How do you find the area of a trapezoid?

Category: 2D Shapes »

Course: Geometry »

 
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The Impolite Quadrilateral

We are asked to use area calculations so much in Geometry that we start to not even think about it. And when it comes to four sided figures, learning about rectangles and squares seemed so intuitive - the area of each is "base times height". Heck, even the parallelogram, which seems like it should misbehave, also has area "base times height" (although in that case we do need to be careful to use the actual standing height of the figure, not the slant height).However, the pesky trapezoid stands out as its own figure with a unique and easy-to-screw-up area formula. Fortunately, however, if you look at the formula the right way, you can not only remember it forever, but also understand the intuition behind it!Here's a standard issue trapezoid:A trapezoid has exactly one pair of parallel lines. We'll call those lines base 1 and base 2 (notated $b_1$ and $b_2$). This is the driver of the difference in area formulas; other more "regular" four sided figures have two bases as well, but in those cases the two bases have the same length. The trapezoid does not. Therefore, we cannot expect the simpler "base times height" formula to work for a trapezoid, since the very definition of "base length" is not the same as other quadrilaterals.What is true, however, is that the "base times height" scheme will work on a trapezoid if we use the average base! Recall that the average of two numbers is computed by taking half of their sum:$$\mathrm{Avg} \,\,\,\, \mathrm{base} = \frac{b_1+b_2}{2}$$The area of a trapezoid cannot be stated simply as "base times height" because it has two different bases. It can, however, be stated as "average base times height".So now you not only have the formula for a trapezoid, but also a way to remember it!