Arc

[Jerry Thompson]

I need to know how to find the arc of a circle segment. As an example if I use a pail to make an arc to make a truck lid I need to know at what bevel to rip the strips for the top.
The othe problem I have is that I am math ignorant. I have had no math education with the expertion of the very basic, e.g. add, subtract, muliply and divide.
So please be gentle with me and use numbers not x, y, a,b, etc.
Thank you.

[mickey cassiba]

Jerry, there are many "cheat sheets" available on line. I'll look through my files and try to find one or two. The best of them allow you to enter known values, and give back an answer. There is a website called "Instructables" that contains many links to helpers for "math challenged" folks such as myself.Google (and Bing as I have been leaning toward more and more) are your friends.

[Andrew Pitonyak]

I am not certain that I can do this without using variable names because you did not supply numbers from which to start. I did this on the back of a piece of paper, but, I will run a few tests to see if they feel correct.


When you say that you want to measure the arc, I assume that you mean something like this:
Circle.jpg
You desire to build an arc that runs along the edge of the circle starting where the horizontal line hits each of the circle. In the picture, I have labeled half of the total width as w and the height from the horizontal line to the top as h. I assume that the radius (r) of the bucket, which means the length of the arrows, is 10".

If I know w and h, but not the radius, I can find r as (w*w + h*h)/(2*h)

I want to know the angle from a red arrow to the black arrow, which is half of the arc. The sin of the angle is given by the half width divided by the radius (w/r). I will call this angle theta.

theta = asin(w/r)

For a simple example where w = h (say the width and height are both 10, then the radius is 10. This means that we have a half circle. In this case, theta = asin(10/10) = asin(1) = 90 degrees. If your calculator comes out with a number more like 1.570796327, then your calculator is in radians mode. Either change your calculator to use degrees, or multiply your answer by 180 and then divide by 3.141592654

let n be the number of boards that you want to use. (say, let n=4, so you want to use 4 boards).

You should cut each board that will connect to another board at an angle of theta / n. In this example, you want four boards so you will cut each board to 90 / 4, which means 22.5 degrees.

Another question is, at what angle to cut the very ends so that they will sit flat if you stand them against a non-angled surface. I believe that this value is

theta * (n-1) / n

So, with n = 4 and theta = 90, the angle is 67.5 degrees.

Now, how long is each board along the outside edge? Use the law of cosines to determine that the out length is sqrt(2*r*r - 2*r*r*cos(2 * theta / n))

In our example, that means that each board is 7.6537" long


What if we let w=6 and h=2 and n = 5? I believe (right or wrong) that r = 10, theta = 36.869897 degrees, the join angle is 7.37 degrees, the flat angle is 29.5 degrees, and the outer length is 2.5669.

Assuming that my math is correct, of course.

[Allen Schrader]

Maybe this will help: http://www.fisherwoodcraft.com/tips.php?TIP=calc

[Dave MacArthur]

erm... circumference of a circular secton is
theta*r
where theta is the angle measured in fractions of pi and r is radius in any units.

Angle of the board edges: take whatever the total angle subtended by the arc is, express it as fractions of pi (a 90 degree angle is pi/2). If you have one board then that is the angle between the two "radial faces", like a 90 degree wedge from a tree trunk. If you lay a board across that subtended arc, and measure the angles from the axis/face of that board, then you just divide by 2. So, board laid across the chord of a 90 degree subtended arc of a circle has side angles cut at pi/4 (also known as 45 degrees) to match up with other boards.

If you cover that arc with two boards, then you divide your answers by 2. If you use 3 boards, divide by 3. Etc.

Here's the real secret: ONLY USE POLAR COORDINATES when dealing with circles, and everything is simple! Rectilinear coordinates and angles expressed as degrees are not the correct system to do circular functions in. Convert your answer to degrees once at the end to set your table saw.

[Brian Vaughn]

Jerry, there's some more complicated solutions being posted here, but the reason is that we don't have any real numbers from you to help...So let's try this.

How big is the diameter of the pail you're tracing around?
Take that number, and multiply it by 3.14 (3 would most likely be close enough here). You now have the circumference of your pail. Let's say the pail is 1 foot in diameter, then 1 x 3.14 = 3.14 feet around.

I'm gonna hope, since you have woodworking skills, that you might have an old fashioned compass around. You need to find the angle of your arc. For instance, if it was half the pail, it'd be 180 degrees. If it was 1/4 of the pail, it'd be 90 degrees. Just make sure you measure that angle from the center of the circle.
Take the angle, divide it by 360, and then multiply it by your circumference.
In our previous case, we'll say the angle is 90 degrees. That means 90/360 = .25, and .25 x 3.14 = .785 feet. That's the length of your arc.

Next, figure out how many strips you want to use for the arc. Divide the angle of your arc by the number of strips, and then divide that number by two. In this case, I'll say we're going to use 10 strips. Since we figured out that our angle was 90 degrees, then 90/10 = 9 degrees. And 9/2 = 4.5 degrees. That's the bevel angle if you will be beveling both sides of the strips. If you're lazy like me, you may only want to bevel one side of each strip. In that case, don't divide by 2. (Ours would have been 9 degrees)

Last thing, you have to figure out how wide to make the strip. We figured out before that we needed .785 feet of curved surface. Assuming that will be the outside surface of the curve you're building, just divide the length by the number of strips. In our case, that would be .785 feet / 10 = .0785 feet. (.942 inches)

So for me to build my 90 degree arc with a 1 foot diameter, I'd need 10 strips, beveled at 9 degrees on one side only (Remember, I'm lazy), and the wide part of the strip would need to be .942 inches wide.

Keep in mind, if you're lazy like me, that means the inside surface will not be smooth, because the beveled edges will be longer than the straight edges. But if all you care about is the outside surface, it'll do you just fine.

[Pat Barry]

Brian, If you only bevel one side at 9 degrees, and leave the other square, then the bevelled piece will not fit properly against the square edge. The bevelled edge will be marginally longer and this will create issues.

To me the best answer is to draw the circular segment to actual size (arc based on the pail for example), then make a parallel inner arc which represents the thickness of your intended boards to be used as the slats, then simply draw perpendicular lines to represent the joint lines at what you think makes the best visual appearance. Draw straight lines to connect the intersections of the arcs and the perpendicular lines. Now all you have to do is lay your protractor on the resulting angle and you have what you need. This will take approx 10 minutes time and give you all the accuracy you need.

[Pat Barry]

Here is a quick sketch to go along with my previous post. I apologize for the non-roundness of the arcs drawn with sketchup.
arc.jpg

[Brian Vaughn]

Pat, you're correct, the length of joining faces will not be the same length, which is why I said "the inside surface will not be smooth, because the beveled edges will be longer than the straight edges". In other words, you can line up the outer corners and make a nice smooth arc on the outside, but the inside will be "stepped" (for lack of a better term) due to the difference in length. By the way, at the 9 degree angle we used before, for a 1/2" thick board, the difference between length of those two faces will be .006 inches. Not a huge deal, but something to be aware of.

[Jerry Thompson]

I am using a hypthetical situation e.g., a curved trunk top. The top would be 16" wide by 24" long. The top of the arc would be 4" high at the center. I would use say 15 boards 4"wide. I can figure out what bevel needs to be cut in the boards but I have no clue as to figuring out the arc.

[Dave MacArthur]

ChordAndSagitta.jpg
C = the "chord", which is the length cutting across a circle between any two points on the circle
h = the "saggita", which is the height from the chord to the top of the circle arc above it
r = the radius of the circle to make that happen

The "ARC" you keep saying is not a correct term, so it's difficult to know what you mean... the only other piece of information is the angle. Well, you can also measure the "arc length".

Using your example of a treasure chest lid:
You know the chord, 16", that's the distance across the top the lid must span
you know the sagitta, or height of the lid, 4".
The radius of circle which will allow a chord to be cut across the arc produced by some angle (alpha) with a chord length of 16 and a sagitta (h) of 4 is:

r = h/2 + (C*C)/(8h)

or with numbers for your example: r = 4/2 + (16*16)/(8*4) = 2+ 256/32 = 10
So, a 10" radius circle. It sounds small, but I just plotted it out and drew the curve, and it's correct.

You can also get the angle that subtends that arc, numerous ways:
a is 1/2 the total angle, lets you have right angles to do sin/cos/tan:

sin a = 8/10, so angle = arcsin (.8)
cos a = 6/10 so angle = arccos (.6)
tan a = 8/6 so angle = arctan (1.33).
all give an angle of 53.13 degrees, so twice that is 106.26

The length around the lid for calculating boards to use is: S= angle in radians * radius.
You can also quickly do it with degrees if you don't have radians by: there are 2pi radians in a circle, so:
106.26 deg/ 360 deg = arclength / 2pi*r (just set up a known equality of ratios and solve for what you want, here it's arclength....)


or: S = (106.26/360)*6.28r = 18.53" (if I'm not mistaken).

You can imagine that if you wanted the height of the lid to be 8", that is exactly half it's width, than a circle with center right there in plane with the top would have an 8" radius to either side and 8" to the top... I've drawn that out for you also.

[Roderick Gentry]

You have a truck that is 16 inches by 24 inches, and the top is 4 inches high, and you hare using 15 4 inch boards? Where?

My feeling about your problem is that it is standard boat building, I just need to understand what you are trying to do a little better. Trig, never needs to come into practical boatbuilding so you don't need to touch math. I am comfortable with trig, but I have shifted to drawing stuff in CAD. It is faster, and more information rich, and very easy. A big complex drawing in CAD is a major skill but drawing circles and squares will get you the info you need and is about the same difficulty as moving your cursor around a document to edit a sentence.

So in 11 above you could draw that curve using a 3 point definition of a circle. To get that. I would use a the rectangle function to draw one 16x4. Then snap the three points to the lower corners of the rectagle, and the midpoint of the upper side. This will give you the full size circle. You can then snap lines to it's center, and extend lines to the arc where the angles/joints will be. You can also output the drawing to a printer if you want to.

The way a boatbuilder would do that curve is with a construction where you draw a line, 16 inch long. then draw a 4 inch circle in the middle of it. The arc of the circle is used to extract heights above the line along the actual arc, and these points are connected with a batten.

Another way is with two sticks that touch the ends of the line, and overlap 4 inches above the center. This is tacked together. This device can then be used to draw the arc with a pencil directly. The sticks need to be at least 16 inches long below the overlap. The pencil is held under the overlap and the device is slid along so it never looses touch with the ends of the line.

See, no numbers.

[Dave MacArthur]

Or, you could do it in the universal language used to think about and handle such problems, math, in about 20 seconds, and never get up from your chair And the answer would still be 10" heh.

I'll admit, I'm interested in this two sticks idea, now I have to go out to the shop and build one...
In all honesty, I think the "correct" approach to design philosophy in this example would be a more hand's on "two sticks" method... sure, if you KNOW that you want an arced trunk top that is 4" high, and 16" wide top, then you can figure out the numbers... but how do you KNOW that's what you want? If you pick a number and solve for it, you're just guessing... I think a better way is to have a device that you can rapidly make various changes to, zip zip zip... it THAT the look I want? No, 3" looks too low... how bout THAT? No, top looks too steep... this? YEAH, that's the look I want right there, and I know it is because I'm actually LOOKING at it.