Equilateral Triangles for Dummy Geometry Students

Equilateral triangle area derivation

There were a lot of reasons why eHow.com became the poster child for content farms gone wild before the Google Panda update started killing them off. One was the incongruity of a college sorority girl turned kindergarten teacher lecturing on butt plugs, the other was that the the site's bullshit to information ratio was far too often astronomical. Here's an example of the latter: take one business major (Mark Kennan, sometimes known as Michael Keenan), ask him an ambiguous question ("How to Calculate an Equilateral Triangle from Height"¹), and watch him flail at it on the so-called Sciencing.com website... as a "physics" question?

Our staffers quickly came to a consensus conclusion: since equilateral triangles have equal sides, it's a reasonably simple process to determine the length of those sides from a known height using the Pythagorean Theorem. Kennan, however, decided that the OQ wanted to know the area; and, in the process, blew right by determining the length of a side.

For the record, to determine the side, consider the following for the equilateral triangle ABC (above):

1. Construct a line BD to represent the height, where D is the midpoint of side AC
2. The length AD is one half the length of  any side
3. The side AB is the hypotenuse of the right triangle ABD, so according to the Pythagorean Theorem, AB² = AD² + BD²
4. Since AD = AB / 2, that equation becomes AB² = BD² + (AB / 2)²
5. Squaring (AB / 2) results in (AB² / 4), which makes the equation AB² = BD² + (AB² / 4). That equation reduces to 4AB² = 4BD² + AB², or 3AB² = 4BD²
6. Solve for AB: AB² = 4BD² / 3, take the square root of both sides and AB =  2BD / (3^0.5)  (2BD divided by the square root of 3).

Using Kennan's example of a triangle with height 14, you end up with an answer of 16.166. If you know trigonometry, it's even easier: AB = BD / sin(60°). And once you have that, you know that the area of your equilateral triangle (formula A = ½AB * BD) = 113.16 -- the same answer Mark got... except here's the critical step in how he got there:

"Divide the height of the triangle by the square root of three times 0.5..." Really? he just plucked that factor – square root of three times 0.5 – out of thin air? and some idiot J-school grad content editor let it pass?  We know where it came from, because we just went through the steps necessary to derive it, but could Mark explain it? No? We didn't think so; and that's why Kennan gets another Dumbass of the Day award.

¹ The original has been deleted by Leaf Group, but can still be accessed using the Wayback machine at archive.org. Its URL was   sciencing.com/calculate-equilateral-triangle-height-8087405.html

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