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I am creating a three-legged stand for a monitor (not really a table, but whatever) and am not sure how to conduct the measurements so that all three legs are accurately spaced in an equilateral triangular shape such that the weight of whatever is sitting on top is evenly distributed over all three legs. How do I go about calculating this?

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Are these legs straight or splayed? What do you mean by "accurately spaced? What weight? Do you need an equilateral triangle or can it be isosceles? Why not just trisect the circle? –  bib Oct 4 '12 at 23:03
    
Equilateral (thanks, Google). Legs will be straight. I guess you can forget weight. I just need even balance and the equilateral triangle leg formation would seem to be the best route. –  oscilatingcretin Oct 4 '12 at 23:09

4 Answers 4

There is a rather simple tool called a protractor which is taylor made for this situation. Here is an example of one:

enter image description here

These are available in various sizes in the school supply section of many retail markets.

Simply draw a line from the center of the table top to the outside perimeter (1). Then use the protractor to measure off an angle of 120 degrees from each direction from the line you have drawn and make two more lines (2) & (3). These will determine the rotational location for the three table legs. Measure from the center of the table along each line an equal distance to locate the center position of the legs (4), (5) & (6).

enter image description here

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Construct a hexagon, discard every other vertex.

Start with a no flexible string (wire works very well) and draw a circle. The center of this circle is where the center of the table's weight should be. The circle should be wide enough that your legs will attach somewhere along it's edge.

Then reposition to where you want one of the legs to be and draw (most of) another circle. Since the wire is the same length, it will pass through the center of table's weight.

Then reposition to where that second circle crosses the first (it will do so in two places) and draw more (partial) circles. They will go through the center of the table and the point where you drew the first off-center circle. Since your string didn't change length, the three points (center of the main circle, and center of the two other circles) are all the same length apart, making a perfect 60 degree / 60 degree / 60 degree triangle.

Then you repeat the process, moving the center of the "next" circle around the edge to where the last circle crossed the "center" circle. You will construct six crossing points on the outer circle, each exactly 60 degrees apart, and if you connected them you would have a perfect hexagon. To make a triangle, just ignore ever other point.

Here's a video of how it's done.

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An equilateral triangular set of legs is easy for a circular top.

Take a pair of dividers (for those who do not know what I mean, this is just the standard school compass used by kids), set to the radius of the circular top.

Step around the perimeter of the table top using the dividers. There will be EXACTLY 6 such steps until you return to the start point if you had set that distance carefully. Mark the steps you take with a pencil around the perimeter. (Adjust the dividers if you do not return to the start point, then redo the steps.)

Once 6 points are located around the perimeter, choose every other mark. Draw a line from the center of the table top to the corresponding mark on the perimeter.

Finally, with that same compass, draw a circle around the center point. Where that circle crosses the radial lines you have just drawn will be the location of your legs, so choose a reasonable radius here. Stability can be increased if the legs splay out a bit, but not too far as you don't want people tripping over them. Rungs that connect the legs can make them more robust.

The process I described above works because a regular hexagon can be inscribed inside a circle such that the vertices lie on the perimeter of the circle. The length of the edges of that hexagon are the same as the circle radius.

enter image description here

Theta = 360/12 = 30 degrees, Sin(30) = 1/2 So the chord length is R.

Edit: You often have the center of the circle from when you made the table top. Some circle cutting jigs will actually leave a hole in the center.

But, if you don't have that mark to use, that compass will come again to your aid. Set that tool to mark a distance larger than the radius of the circle. With one point at the perimeter of the circle, draw an arc across the rough center of the top. Now repeat, but from the opposite side of the table. Those arcs will cross at two points. Draw a straight line between them.

Now, rotate the entire process by approximately 90 degrees, drawing a second line between a pair of intersections of circular arcs. The two lines we have drawn will intersect at the center of the circle.

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Presumably, one has already found the center in order to cut the table top. –  Chris Cudmore Oct 5 '12 at 14:23
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Finding the center of the circle questions belong on math.stackexchange.com ;) –  The Evil Greebo Oct 5 '12 at 14:54

An easier method to using dividers is to get a piece of string, wrap it around the circumference of the table surface, and mark the and where the string meets itself. Measure the length, divide by 3, then mark off the thirds on the string. Put the string back on and use the third marks to find the leg positions. It's not as exacting as @woodchips method but it's faster, you don't need special measuring tools, and it's probably good enough for your purposes.

Of course you could just eyeball it, you'd be surprised how exact you can get it that way.

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I like making lamps of wood. Some of mine are pole lamps, meant to stand alone on a tripod base. One of the lamps I made just looks wrong. The legs are spaced at the wrong angle, not quite 120 degrees apart, and you can easily see the error. Clearly I mis-marked the leg positions, and just drilled away where I had marked. The moral is measure once, cut twice. Luckily, this lamp just stands in an upstairs room, not intended for sale, or it would be only good for firewood. This is how one learns the hard way to measure these things carefully. The eyes can lie. –  user558 Oct 5 '12 at 11:28
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Having made that comment, I will assert that I kept the lamp, and am still using it. It simply bugs me every time I look at it. –  user558 Oct 5 '12 at 11:32
    
It all depends on how exacting you want to be, and who will be seeing it. In extreme cases I'd make a cardboard cutout of the tabletop, be obsessive with my measurements, and then overlay the cardboard on the tabletop for drilling and cutting. Other times I'd just wing it. To me it depends on the audience. –  GdD Oct 5 '12 at 11:47
    
That was the point of my anecdote. For something that I will use for myself, as long as I can accept an imperfection, a small mistake is nothing more than a curiosity that I should laugh at. That small mistake would make the product unsellable if that were my goal though. So know yourself, and know your goals. –  user558 Oct 5 '12 at 12:26
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I'd remove the line about "eyeballing it" your string method is worth the answer alone. –  ChrisF Oct 5 '12 at 15:42

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