Quadrilateral Area for Dummies

irregular quadrilateral

If you've spent any time with a middle-school math student lately, you have probably been introduced to the various formulas for areas of polygons. In particular, there are different formulas for four-sided polygons (quadrilaterals) depending on the shape in question: rectangle, parallelogram, or trapezoid. But what if the shape is none of the above? In that case, you want to know "How to Calculate Area for an Uneven Quadrilateral." We suggest, however, that you not look to Beverlee Brick and her Sciencing.com post for an answer...

The three types of quadrilaterals we've already mentioned are even quadrilaterals; meaning that they have one pair (trapezoid) or two pairs (rectangle, parallelogram) of parallel sides. Other shapes –"diamond," "kite," rhombus – are also even quadrilaterals. There are simple formulas for the the area of an even quadrilateral, but figuring out the area of an uneven version – one with no two sides the same length, to be pedantic – is more complex.
Brick actually said something along that line. Where she got messed up was in her choice of uneven quadrilaterals, beginning with her instructions to calculate,

"Area of a Parallelogram: Draw a rectangle using the two straight lines as the sides. Calculate the areas of the rectangle and the two triangles formed outside the rectangle. Find the area by adding the areas of all three shapes."

We guess that, if Beverlee had taken the time to do a modicum of research, she would have learned that there's a much easier way to calculate the area of a parallelogram: you use the formula Area = base * height. It works because the sloping sides of a parallelogram are parallel. Beverlee might have confused a parallelogram with a trapezoid, but even that shape has a simple formula: the area is equal to the height times the average length of the parallel sides. Duh...

Brick got closer with her discussion of another shape:

"Area of a Kite (Rhombus): Draw a two straight lines to connect the opposite corners of the kite to each other. This will divide the kite into four right triangles. The area of the rhombus is the total area of all four triangles. "

Well, again, there's a simple formula for a rhombus (which, by the way includes more than just a "kite" shape): The area of a rhombus is half the product of the two diagonals. In fact, if you were to expand on Beverlee's "solution," you'd get to that formula; but only if you understood a) geometry and b) simple math. Guess that leaves out Brick.
Still, she's only discussed "even" quadrilaterals. The answer to the OQ's quandary is hidden in Beverlee's discussion of a "kite": you draw a single diagonal across your quadrilateral, calculate the areas of the two triangles it creates, and add them.

If you're into higher math you can calculate the area of an irregular quadrilateral given the lengths of the sides, one or more of the angles between the sides, the angle between the diagonals, or the lengths of the diagonals. Brick, however, seems uncomfortable even with "lower math," hence her fourth Dumbass of the Day award.

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