*Somewhere* there was an online calc or website that was able to figure out the length of the segments in creating a circle of "X" inches in diameter. I am sure that many of you math types will guffaw, so I'll take my lumps, but I'd like a better way than laying it out on paper and measuring.
I was showing one of the students how to lay out an 8 and also a 12 sided circle, 12" in diameter, with 3" wide stock. Our method worked okay and all, I just know that there's an easier way to do it.
The general formula is that arc length is equal to radius times included angle (in radians). If you're after the chord length, just multiply the diameter by the sine of half the included angle.
The length you want is simply "double the sine of the half-angle". ("sine" takes a trig table, or a calculator.)
Here's _all_ the gory details of what and how:
Obviously the paper-and-pencil method is to draw a circle, draw the required number of radius lines, and "measure" the straight-line distance between the end-points of two adjacent radii.
Now, consider those two adjacent radii, and the portion of the circle that connects them.
Draw the chord (the straight line that you want) that connects the ends of the radii.
Draw a _third_ radius line that goes down the middle of the piece.
Note that it will _always_ intersect the chord at a 90 degree angle.
And that the angle between that new line, and either of the 'edge' lines is exactly half the angle of the whole.
These relationships are true *regardless* of how many sides the polygon you're constructing will have.
Now, considering one of those _right_triangles_, you know the length of the hypotenuse, *and* the angle at the circle center.
This is everything you need to know to _calculate_ dimensions.
the length of the chord is, obviously, twice the length of the side of the right triangle.
the length of the side of the right triangle is computed as the 'sine of the angle opposite it, multiplied by the length of the hypotenuse'.
This does require a 'trig table', or trig functions on a calculator.
Now we've already established that the 'opposite' angle of the right-triangle is 1/2 the angle of the full chord. so we just calculate
(sin(1/2 angle)*hypotenuse)*2
commonly described as "double the sine of the half-angle".
e.g. for an 8-sided 12" diameter circle:
a single side subtends 45 degrees so half that angle is 22.5 degrees for 12" diameter, the radius is 6"
Thus: sin(22.5 degrees)*6"*2 (my handy-dandy calculator says that "sin(22.5 degrees)" = 0.38268) = 4.592 inches (or approximately 4 19/32")
The MS-Windows "calculator" (Start->Programs->Accessories->Calculator) does trig functions, if you click on the 'View' button and select "Scientific'.
"Conveniently", it takes angle input in 'degrees', by default. you don't have to go through the nonsense of converting to 'radians'.
To 'confuse' people, you can just multiply the 'half angle sine' by the diameter.
On the other hand, if your scientific calculator has a "pi" button, you can start with 2*pi instead of 360 and work in radians from the git-go without any conversions.
(I'm picking this nit because I'm aware that John is also working to develop expertise with a programmable woodworking tool whose control software trig functions use radian measure only.)
Some people do the 'trivial 'steps in their head (e.g. 8 sides == 45 degrees,
12 sides == 30 degrees) first, and go to the calculator only for the messy parts. It tends to be faster that way. and you have a 'feel' for the reasonableness of what you're dealing with.
Radians "make sense" when they're expressed in algebraic terms -- e.g. "pi/4"; but as numerical quantities, they leave a lot to be desired insofar as 'intuitive', shall we say, interpretation goes.
Me, I figure that anybody who knows what radians are, and chooses to use them, would have no trouble figuring out what to do, given that the Windows calculator (in scientific mode) has radio buttons for using degrees/radians/ gradians, _and_ a PI button that carries it out to more than the 20 decimal places that I have memorized.
Is this a teaching opportunity to bring some math into the class?
Basic math: Figure degrees in included arc Figure circumference of circle Figure miter angle at end of boards Figure 1/8 or 1/12 of circumference which would give you arc distance, or rough length of the wood side.
I think you would need to use trig to figure the chord distance.
Don't know if you need more problems to deal with or not.
Sad to say, many highschoolers can't even manage fractions properly. I can still hear a faint echo of "look out, the old man's teaching Physics at the lathe again." It may seem the most natural thing in the world to give an explanation, but in a world of magic, where the tube makes anything possible, few are interested. Of course, probably a quarter read so poorly that "look it up" isn't an answer, either.
HS students short on geometry chops are nothing new. They were not ensuring students like myself (class of '71) "got it" even back then. BTW, this is the first time in my life I ever needed it.
Could it be that most^H^H^H^H a great many HS teachers don't have much of a clue (or don't bother to communicate) the value of the subject they teach or how it applies to the world outside the classroom? High school graduates who don't know geometry or trig are generally able to say "I don't know," and then go dig out what they need - a very different situation from those who didn't learn anything in history classes and then confidently go to the polls and ensure that we repeat past mistakes.
I didn't have much use for geometry or trig myself until I got involved with trying to write optimizing CNC program generators - and found myself up to my eyebrows in the stuff (and, most remarkably, enjoying the challenges in solving the problems that popped up!)
As I think about it, your method of "draw it full size and measure it" isn't all that bad. Gives the students a quick method of solving a problem using their native wit, instead of spending a bunch of skull sweat on abstract concepts.
I actually think it is a wiser way to go than trusting in a "magic" computer program that spits out the answers without the user understanding the process.
A local shop machines stone for buildings. When they had to do curved sections they would calculate the heck out of it, then get a piece of string and chalk and mark the radius on the shop floor to double check the fit. Only after CNC machines and CAD did they find they could skip this step.
"Walt Cheever" wrote in news:5S4be.18532$NU4.8092@attbi_s22:
After I lay it out on the floor, I generally change something to get it to 'look right' anyway. Most of the numbers are guesses, based on getting to 'looks right'.
Of course, gluing up a 13" circle to fit on a lathe with a 12" swing is a good way to look the idiot in front of a couple of students in shop class.
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