Arc Length for Dummy Geometry Students

arc length as function of internal angle

It's not unusual for our research team members to run across bizarre questions in former Demand Media sites such as eHow.com, Hunker, Techwalla, and Sciencing. Sometimes the only thing more bizarre than the questions – which have been harvested from real internet searches – are the answers to those questions concocted by the sites' contributors. Take, for instance, Amy Dombrower: Amy was saddled with the loopy-sounding question, "How to Convert Degrees into Inches or Millimeters."¹ The plucky J-school graduate gave it her all, such as it is, for Sciencing.com; but they finally realized her answer was bullbleep and handed it to one of their house crap-fixers to rewrite...

It's a rather dumb question without context, but Amy came up with her own idea; which is to assume that the question is about how to determine the length of an arc of a circle if you know the arc's internal angle, which we'll call θ. We figured it wasn't all that difficult: you know what fraction of the perimeter the arc covers (it's θ / 360) and that the perimeter is 2π times the radius R, so it should be easy to determine. With a little math, that relationship reduces to θ / 360 * 2πR, or  θπR / 180. Easy-peezy...

After a doofus explanation of the problem:

"In geometry, a degree is the unit of angle measurement, equal to 1/360 of a circle. When measuring angles from the center of circle, degrees can also be expressed as radians. The radian measure of an angle is the ratio between the length of the arc opposite the angle and the radius of the circle. An arc is a portion of the circumference of a circle, so the arc length is defined as the length of that portion. By converting an angle’s degree measure to radian measure, you can find the arc length opposite a given angle, measured in a unit of length such as inches or millimeters." Dombrower embarked on a tortured set of steps to explain how to

1. Convert the angle to radians (example: 50 degrees = 5π/18 or 0.8727 radians)
2. Determine the radius (example: 10 inches)
3. Convert the radian measure to arc length s (s = 0.8727 * 10 inches = 8.2727 inches)
4. Convert the answer to millimeters (8.727*25.4 = 221.666 mm)

All of which required more than 250 words and introduced the concept of radians, which we suspect completely blew the J-school grad's tiny little mind (not to mention exposing her ignorance of the significant digits concept).

We submit that our answer, which gets the same result, is a damned sight simpler, Why? Because we have people on staff who know how to think through the problem and aren't afraid of performing a little mathematical manipulation. That's as opposed to our Dumbass of the Day, who had to look it up and very likely didn't understand what she found. Duh... Oh, and that "crap-fixer"? Chris just tightened Dombrower's verbiage and still had to go to radians...

¹  Leaf Group actually managed to find someone to rewrite this dreck in a more-or-less cogent form, but Dombrower's original can still be found using the Wayback machine at archive.org. Its URL was   ehow.com/how_6692856_convert-degrees-inches-mm.html

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