Radius from Chord, the Dummy Version

Calculate a circle's radius from the length
and height of a chord

It's been a while since our research staffers passed along an example of just how poorly the average owner of a liberal arts degree understands mathematics beyond simple addition and subtraction; although some have pretty sketchy skills there, too! One such self-appointed expert we've seen several times plying the freelance mathematician trade, albeit badly, is eHow.com's Charlotte Johnson, already a five-time DotD with four awards in mathematics! Charlotte's chosen task this time is to explain "How to Find the Radius of a Circle From a Chord," a procedure of intermediate level that's covered in eighth-grade geometry classes. (Note: the post has been moved to Sciencing.com).

The real answer? Given the length of the chord l and the height from the chord's midpoint to the arc h, the formula for the radius r is,

r = ((l²/4) + h²)/2h
And it's as simple as that. So let's see how Charlotte approached it in the DMS format: Johnson began, as eHow contributors are required, with an introduction; a passage that suggests she was already out of her mathematical depth:

"Dealing with parts of a circle, such as radius and chord, are tasks that you may face in high school and college trigonometry courses."

Well, maybe... but we suspect you're more likely to run across the problem in a geometry course... Charlotte then laid out the steps, which (per DMS requirements) number five. We've reproduced them in their entirety, including her example (a chord of length 4 with a height of 2):

Multiply the height of the chord times four. For instance, if the height is two, multiply two times four to get eight. Square the length of the chord. If the length is four, for example, multiply four times four to get 16. Divide your answer from Step 2 by your answer from Step 1. In this example, 16 divided by eight is two. Add the height of the chord to your answer from Step 3. For example, two plus two equals four. Divide your answer from Step 4 by two to find the radius. Therefore in this instance, four divided by two equals two. The radius in this example is equal to two.

The first thing we noticed is that Johnson's chord is very short in comparison to the height. In fact, the chord in her example is, by definition, a diameter: the length of the chord is equal to exactly twice its height! Bad example, Charlotte, bad example!

When all is said and done, Johnson's steps produce a version of the right answer, which looks like this:

r = ((l² / 4h) + h) / 2

...although we suspect that's an accident. Right or wrong, however; she started her explanation in the middle of the formula ("calculate 4 times h") -- a starting point that confused our house mathematician at first -- and then used a truly crappy example. It's just one more case of a liberal arts major attempting to explain a math problem she didn't understand: a classic Dumbass of the Day move!

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