Question is are circles better than squares for determining accuracy on your CNC?
Depends on what kind of accuracy you are talking about. The method avid recommends is drilling four points on scrap placed in each corner of your bed and checking the diagonals, this is great for checking skew.
@jjneeb has a great video on linearity and backlash calibration. https://www.youtube.com/watch?v=p9nsaW7acuQ
I used a jig I made with a camera sensor for improving the level and linearity of my rails after watching this video: https://www.youtube.com/watch?v=hnHjrz_inQU
I use a surface plate that has been leveled to the machine xy axes and a square for checking Z-axis tram:
And the classic dial indicator method for spindle tram:
Either one could tell you X and Y dimensional accuracy, as well as squareness. The square could tell you if your X and Y rails are straight.
There are other aspects of accuracy that these won’t tell you.
What aspect of accuracy are you specifically trying to measure?
Well I was looking for an easy inexpensive way to check the accuracy of my machine. Without using dial indicator. So I googled and knowing that Google is not always right. It said that cutting a circle will check the x and y and backlash making that the best way.
Is that true, what’s your opinions?
Welcome back Rich it’s been a while 
How did you do that did you use a laser router bit?
60 or 45 degree V bit at around 1/32" doc. The arrows indicate direction of cut.
I did a second sheet using colored marking pens where I traced over the same patterns twice to look for discrepancies and found some. (don’t have pics of that though) – Still, not bad for a 13 year old chain drive machine.
Well, yes and no.
First and foremost, you are always limited by the measurement metrology. So if you are using a tape measure or just a long precision rule, you are limited by its accuracy, as well as the resolution of the measure’s graduations, and your ability to see it. The last two are normally going to be the first order effects, especially if you have old eyes like mine.
To optimize the measurement error, you want to make your object size large so the error to measurement ratio is as small as possible. That is why Eric’s method in this video https://www.youtube.com/watch?v=qz6VrsEq_Xc&t=21s is so good for the X-Y square. It eliminates the error at the far end because you are hooking the tape on a pin, it is a relative measurement between diagonals (so the absolute accuracy of the tape measure is irrelavent), and it uses the maximum size of the table so the error relative to the overall measurement is as small as possible. This is a pretty accurate test. It also makes smart use of scrap mdf or particle board pieces instead of a whole sheet.
So I think you are saying you basically want to use a similar method for X and Y accuracy? If so, I would use that same exact corner method that Eric describes in that video, but then after you get done with the squaring part, measure between the pins at both sides in the X and Y directions. Make your hole pattern something that is easy to measure. For example, if you have a 4x8 machine, and are using 0.25" steel dowels, then make the hole pattern 47.75" x 95.75". That way you should get exactly 48 x 96 on the outside of the dowels.
Since a tape measure’s hook can have some error if you’ve dropped it on the ground a couple times, have someone help you to double check it to see if it is damaged. Measure it with the hook to 96", and then measure from the 1" to the 97" mark to see if you get the same measurement. This is important for measuring the linearity error because unlike the squaring test, it is not a relative measurement and the overall accuracy of each measurement matters.
You really only need to measure X at the back or the front, since any error on the X axis should be replicated anywhere on the table. However, Y can have its own error on either side since there is an independent drive train on each side, so check them both out at the far left and right extremes of X travel.
Now if you want to check the straightness of your X axis (essentially, does your gantry have a bow in it), or your Y axis (does your table have a left or right bow between front and back), you can drill a hole for those 1/4" dowels at 0, 24, and 48 inches on X and 0, 48, and 96 on Y, put dowels in all holes, and then put a straight edge or string line down the line (try it on both sides of the dowels to be sure) to see if there is a bow.
Lastly, to test backlash on the cheap, you can drill holes from two directions and insert the dowel to see if it made an oval or not. So if you watch this video https://www.youtube.com/watch?v=viJMIqKCzBI I explain the theory of coming at a coordinate from two sides to make any backlash show up in your actual position. But instead of using the caliper like I did, you can just drill a hole at the same exact coordinate. So for example, write a Gcode program that drills a hole (straight down, not a pocket hole) at 0, 0, then does a rapid move to 0,4 and drills a hole, then rapid to 0,8, rapid back to 0,4 and drill the hole again. Then insert a steel dowel in both holes. The first hole is what no backlash should look like, a nice even fit all the way around. The second hole that was drilled twice should be the same if there was little or no backlash. However, if there is backlash you will see some extra slop in the fit on the axis that was traversed up and back. I would guess you could catch backlash error of 2-3 thousandths at best with this method. Definitely not as good as a caliper, but honestly good enough for most any woodworking.
The Z axis is pretty hard to do without a caliper because of its short travel (any measurement error is relatively large wrt travel). You can do a gross check on it by using a large slot cutter and slicing sideways into a block of wood at a low and a high Z height and measuring the distance between slots. However, this is going to be a fairly gross check, maybe on the order of 50 thousands per foot of error at best using a tape measure.
Hope that helps.
Thank you Jim
I just drill three holes sized for a tight fit on some metal pins, put them in, then check square with a large (woodpeckers) square laid against the pins.