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