Here are some clever, old school layout tricks that you can use for DIY projects. Perhaps you're helping your child with a school project and you want to draw an ellipse that's too big to print. Or you want to drill a few holes centered on the width of a bunch of 2 x 4's, and you don't want to fuss with measuring and drawing a center line on each and every board. Or you might be remodeling the kitchen and dining room and want a nice gentle arch above the doorway, not an arch whose radius is equal to one-half the width of the door--which, though easy to create, might extend past the ceiling. It's not easy to figure out where the center of the desired arc should be located, unless you use one of the two simple methods provided below. How do you divide a 12" work piece into 9 equal segments without math? How do you divide a circle into any number of pie slices without a lot of math? Read on.
Making an Ellipse
An ellipse is a pleasing shape, and of course it's straightforward to draw one on the computer, but it's quick, easy, and dare I say, fun, to create one with a piece of string, some nails, and a pencil. Put two nails or tacks into whatever it is you need to draw an ellipse on. Tie a string into a loop that can be looped around the nails with a bit of slack left over--experiment with the amount of slack and the distance between the nails. Now insert a pencil into the loop of string and begin moving it in an elliptical path by keeping the loop of string stretched tight as you move the pencil. See sketch.
Credit: Brent Farwick 5-3-2014
Making a Center Finder
Suppose you want to drill two or three holes along the centerline of the broad side of a 2 x 4. You could grab your tape measure and measure half the width, mark it, then move down the board and do it all over again, until you've made all the marks you need--that's a lot of fuss. What if you've forgotten that that a 2 x 4 is not actually 2" x 4" (it's 1.5" x 3.5") or you're not quick at mental math? Make a center finding jig--a handy little tool that couldn't be simpler to make. Find a scrap of wood (use the sketch as a guide to the necessary length). Drive a couple of nails through it, or carefully drill a couple of holes for a snug fit with two small diameter bolts. Draw a line directly between them, and at the midpoint of that line drill a hole to fit whatever marking tool you intend to use--a pencil, a tiny nail, or a nail punch. To use the tool, simply lay it over the board to be marked, and twist it so the pins are hard up against the sides of the board, thus centering the middle hole. Mark. Pat yourself on the back for being so clever.Credit: drawing by Brent Farwick 5-2-2014
Laying Out X Equal Segments on a Given Length
Suppose you wanted to layout 9 equal segments on an 12" long workpiece. So you reach for the calculator only to find out that each segment will be approximately 1.33" long. Well, that's not handy! If your tape measure or ruler doesn't have the desired fractions for such a task, or if it's too hard to mentally add them up on the fly, there's a handy little trick to take most of the math out of the equation. Suppose your segments are a just a bit less than some handy measurement on your ruler. Simply angle your ruler as shown below, mark the segment divisions at the ruler's edge, and use a square to transfer the marks to your work. This technique is not limited to things that fit on your table or workbench--the driveway or floor can serve as your bench for larger layout work.Credit: original sketch by Brent Farwick 5-3-2014
Finding the Center of Any Arc
For a math-free method, see the last paragraph of this section.
Suppose you would like the top of a certain doorway to be an arc (portion of a circle) rather than just a horizontal line. Your first reaction might be to use ½ the width of the door as the radius of your arc. If so, it will be very easy to lay out that arc above the existing doorway. But if you prefer a shallower (less tall) arc, then it's not's so obvious. What is the radius of such an arc? It's certainly going to be more than ½ the width of the door, and it's not immediately obvious how to find the radius if you start with the the maximum height of the arch above the door as your controlling dimension. The technique below will allow you to determine the center of any arc (segment of a circle) with at most a ruler and a square (or plumb bob).
Consider the intersecting lines in the sketches below. In both cases, a x b = c x d. Indeed the product of the component parts of any two such intersecting lines are equal. The little squiggly on the bottom of d just means that d has been cut off for convenience, but that it really goes all the way to the other side of the imaginary circle, i.e. c-d is a chord. The lines drawn in the arc sketches below are called 'chords'. A chord is any line that stretches from one point on a circle (or arc) to another point on that circle. It doesn't take much imagination to see that doorway we've been talking about in the lower sketch. If we follow the famous Feynman Algorithm:
- Write down the problem.
- Think really hard.
- Write down the answer.
Then we can see that to find d all we need to do is:
- Multiply a times b.
- Divide result by c.
- Add c to the result.
- Divide the result by 2.
If you're the 'mathy' type, use the formula (below right).
Either way, the result is the radius r of your arc.Credit: Brent Farwick 5-2-2014
The reason this works is because the lower sketch is a special condition: The line c-d is:
- A diameter (even though we don't know what size the circle is yet).
- It's at a right angle to a-b.
- It's drawn through the midpoint of a-b, i.e. it bisects a-b.
- It's a chord, i.e. it runs from one point on the circle to another point on the circle.
So to recap, the width of the existing doorway is the first thing we measure, and a and b are each equal to half of it. Then we decide how high above the existing doorway we want the top of our arch to be--that's our c dimension. We multiply a times b, divide by c, add the result to c and divide it all by 2. Voila! We have the radius of our circle. Now go build that arched doorway.
There is another, even simpler method for laying out an arch. Find a flexible, springy strip of steel about an inch or two wide, and a foot or two longer than the arc to be marked out. Holding one end of this flexible steel strip in each hand, raise it to the top of the existing doorway and simply spring (flex) it into the desired shape. Have a helper trace out the arc while you hold the strip in place. Couldn't be easier.
Divide Circumference of a Circle Into Equal Segments
Suppose you want to distribute 13 objects evenly around the circumference of a circle, or that you just want to divide a circle in 13 equal pie slices. How do you do it? If you do the math the central angle of each segment is going to be 27.69 degrees, not exactly helpful unless you happen to have a quality transit laying around. You could use a cad program to do the job for small circles. If the circular workpiece you are using can be physically picked up and rolled, there's another way. See the instructions and diagram below, which assume you're doing this on the floor (but you can do it wherever it suits your needs).
- Make a radius mark on your circle.
- Stand circle on its edge with radius mark pointing straight down.
- Rotate circle one full revolution along a straight path, marking both its start and end points.
- Draw a line between the start and end points.
- Divide the line drawn on the floor into however many equal segments you want--remember that you'll need one more line on the floor than the number of segments. If this confuses you, remember that it takes two lines to mark out one segment.
- Roll the circle back over this newly marked out line, transferring the marks on the floor onto the edge of the circle as you go.
- Pat yourself on the back once more. You truly are very clever.
Credit: by Brent Farwick 5-4-2014
You don't actually have to make a radius mark (as shown in the diagram) if you aren't looking to be quite so accurate. You can just make a small mark right at the edge of the circle. The radius mark (if used) can be checked with a square or a level if more accuracy is desired.
Laying Out Quasi-Parabolic Curves
Suppose you want a nice curve at the edge of some hardwood flooring which transitions into a carpeted area. The technique here is borrowed from string art and many crafters will be familiar with this old chestnut. But it should not be overlooked as a technique for laying out pleasing transitions using pencil lines instead of string. A simple corner transition is shown below (in three stages of completeness) to demonstrate how successive lines are drawn. The net result is a nice quasi-quadratic curve that can be smoothed out with a French Curve (a drafting tool) if desired--but with enough divisions that shouldn't be necessary. The steps are:
- Layout the extent of the transition in both directions.
- Divide each of the above into an equal number of segments, that is, if you choose to have 9 segments on one side, make sure you have 9 segments on the other side.
- Draw a line from the 1st mark on one side to the last mark on the other side.
- Next draw a line from the 2nd mark on one side to the next-to-last mark on the other side.
- Lather, rinse, repeat.
Credit: original sketch by Brent Farwick 5-5-2014
That's All For Now
I hope you've encountered a new layout trick or two here. I make no claim of originality. The origins of these old, but very useful techniques are lost to time.