Black Box Calculation for Dummies

driveway with small radius entrance

You know that old saying that goes, "Give a man a fish..."? Yeah, that one. well, today we're gonna meet a freelancer who hands out fish but doesn't teach people how to fish. We have a pretty good idea why she doesn't: she doesn't know, herself. She's ten-time DotD Charlotte Johnson, who thought she could bullshit her way through the eHow.com question, "How to Figure Square Footage for a Driveway With Radius Entrances" (now at GardenGuides.com). Char found an answer, of sorts, but it's a pretty safe bet she didn't know why it worked or, more importantly, under what circumstances her answer would – and wouldn't – work.

Char started, as required by the eHow overlords, with an introduction of sorts:

Figuring the square footage of a driveway is an important task if you are planning to pave or enlarge your driveway."

That's pretty much in the "Duh!" category. She's also right that, while it's easy to calculate the area of your basic rectangle,

"If your driveway has radius entrances, you will need to apply the appropriate mathematical formulas to determine the total square footage of the driveway."

It's when Char gets to the "appropriate mathematical formulas" part that she cranks up the old black box. Even an idiot (well, most of them, anyway) can tell you how to calculate the area of the straight part of the driveway, it's those pesky little curves at the entrance that make things tough. According to Johnson, however, all you need to do is:

• "Measure the edge of the driveway that connects the side of the rectangular portion to the end of the outward curve. This edge will be flush with the road." – Not a particularly accurate description, but with some puzzling, you can figure out what she means.
•  ("Square the measurement from [the previous step]." – Ummm, maybe. Maybe not.
• "Multiply this figure times a constant of 0.2145." – And here is the point at which Charlotte plops down her fancy black box.

If you're reasonably comfortable with simple geometry, you can – with some thought – figure out where Charlotte's magic constant came from. It goes like this:

Assume that the section of the driveway whose area you're calculating is exactly one-fourth of a circle. A circle of diameter X covers 78.55% of the area of a square with side X; a quarter of that circle covers the same percentage of the area of a quarter of the square. Charlotte's black box calculation simply subtracts the area of the quarter circle from the quarter square.

What's the problem? Well, there are several; but the first is that, for Johnson's black box to be accurate, the radius corner must be precisely a quarter circle. It it's not, well, that black box goes up in smoke. 

An honest freelancer who understood the question and the limitations of this method should have warned people of those limitations. It's a safe bet that our Dumbass of the Day, however, did not understand those limitations. Hence, award number eleven (award number eight in mathematics!)

MM - GEOMETRY