Need Math Help... Checking for Square

[Kevin Godshall]

The panels I have been cutting are measuring square on opposite corners, (with framing square), but out by about 1/8 inch in 4 ft when using the square on the other corners. If I throw a tape corner to corner, my measurements are the same.

OK, math time. Using Pythagorean Theorem (c2=a2+b2), I did some figuring on paper. Taking a model of a 2ft by 3ft square panel, and converting it to 8th's of an inch, I have a piece that is 192/8ths by 288/8ths. Figuring for the hypotenause, I get 346.13/8ths.

Now, taking the same piece and recalculating it, this time taking 1/8 inch off the short side (191/8ths instead of 192) and calculating, I get 345.57/8ths.

Flipping the numbers and this time taking 1/8 inch off the long side (287/8ths instead of 288), I get a hypotenause of 345.3.

Converting 8ths back to inches, it is 43.19" vs 43.16". There is no way I would discover this on my tape measure going corner to corner.

Isn't it true that Pythagoras is only working with a known square angle to start with? How does this affect tolerances in measuring corner to corner to square? Why am I so obsessed with this right now, I'm not getting any work in the shop done? Why is the LOML throwing all my stuff into the yard?

Maybe I'm overthinking this, but till I get it straight in my head, I can't begin to work on why my panels all come out 1/8 out of square with 2 corners square.

Someone, please save me from myself!!!!!!!!

[Brian Kent]

The Egyptians made it simple. If two sides are 3 units and 4 units, then a 5 unit hypotenuse means it is a perfect 90° right triangle.

So measure from any corner you want to check 3' and 4' If it's 5' between those 2 points the corner is square. If not, you found your problem.

[Kevin Godshall]

Hopefully, that is what my framing square is doing for me. I have used 3-4-5 methods on other projects, usually larger in scope.

I know my miter sled is on. I always get a square edge to the edge I used against the sled. My fence must be out just a bit.

My concern extends to the doubts I'm having on using corner to corner measurements as a basis for building boxes, cabinet carcases, etc. Apparently, being the standard for so long and so widely accepted, the tolerance must be OK..... maybe I need a new brain.

[Joe Jensen]

Hopefully, that is what my framing square is doing for me. I have used 3-4-5 methods on other projects, usually larger in scope.

I know my miter sled is on. I always get a square edge to the edge I used against the sled. My fence must be out just a bit.

My concern extends to the doubts I'm having on using corner to corner measurements as a basis for building boxes, cabinet carcases, etc. Apparently, being the standard for so long and so widely accepted, the tolerance must be OK..... maybe I need a new brain.

Boxes are easy. Make sure the corner to corner measurements are the same. You don't need to know the number, just make sure they are the same. If they are then it's square. In my experience making the corner to corner measurements identical is the most sure way to have a square carcase.

[Jason Roehl]

You're mixing techniques, Kevin.

Using the simplified version of the Pythagorean Theorem (I don't recall the full formula for solving without a right angle), you can check ONE corner for square. Don't worry about the actual lengths of the sides adjacent to that corner, just measure out 3 units on one side and 4 on the other, then there should be 5 units between those two points. This is generally more useful on a larger scale when using tape measures (such as squaring up the framed corner of a house. There are other combos, too, such as 5, 12, 13, or any multiple of 3, 4, 5, like 6, 8, 10.

When you're talking about checking a rectangular panel or carcass (bookcase, cabinet, etc.) for square, you are measuring the diagonals against each other to check for square. If parallel sides are of equal length, then for there to be a rectangle, the diagonals must also be equal. If the diagonals are not equal, you have a parallelogram. If it's a carcass, you would put a clamp on the long diagonal to squeeze it until the diagonals are equal. With a parallelogram panel, you find a better method for cutting it...

[Kevin Godshall]

This is where I ran into my problem.

I was cutting a top for a chest I'm making. Ripped both sides to width, then crosscut the ends to length. I threw a tape over the corners, Yep, right on. Put it on the chest and "What the........??" Checked the chest it was OK. Checked the diagonals on the top, it's OK.

Finally gave up on the tape, and went to the big framing square, Yep, chest is OK. Oh my, top is out 1/8th here and 1/8th there.

At that point, I put down the tools and got out the paper, pencil and calculator.

I have 2 good square crosscuts, because I flipped the panel each time when putting on the sled. So the crosscut is square to whichever long side I used in the sled.

So now my question is......... am I happy to use diagonals as my standard for checking square, or do I need to find a better method?

[Kevin Godshall]

LOL Jason, as I posted last response I read the quote under your siggy.

I think that's me right now.............

[Wes Grass]

Have you checked your framing square?

And you should be able to easily see a 32nd on a tape, if you're using the 1" mark as a 'zero' reference. If you're hooking the tip on corners, then forget it.

[Kevin Godshall]

Framing square is good, but its not big enough to measure diagonals on the panel so I had to go to the tape measure.

Looking over the calculations, 1/8 inch off in a 3 foot length is .03" (compared to what a square panel of the same size would be). I can't detect that on a tape measure.

[Jason Beam]

Sounds to me like your framing square is off ... which would explain why you've got two corners that it fits nicely into and two that has gaps ...

EDIT: The reason I made the above statement was that I thought it was impossible to get two square corners and two unsquare corners in a single piece, but it dawns on me there IS one way to get this situation. Illustrated below with an exaggerated drawing:

115927

[Larry Edgerton]

If your opposing sides are the same length, and the diagonals are the same you have square corners.

To check your square find a known strait edge, like a piece of plywood, draw a perpendicular line, flip the square, draw another line starting at the same point, the difference is twice what your square is off.

If it is a steel or aluminum framing square you can fix it with a hammer.

[Brian Walter]

Kevin,

I think your understanding is pretty good. The problem is as you are discovering, diagonal measurements don't provide the degree of precision you are looking for. For example, taking the same 2' x 3' rectangle from your example, if it is square, the diagonal would be 3'-7 9/64", if you hold the 2' side stationary and push the 3' side over so that it is off by 1/8" from the square position, the diagonal measurements will be 3'-7 11/32" & 3'-7 6/32". In other words, the difference is only 5/32" or just under just under 3/16". So depending on how accurately you measure, you might not notice the difference between the measurements.

The diagonal measure technique only works if the opposite sides are the same length. For example, taking your 2' x 3' rectangle, if one of the 2' sides is say only 20" (extreme I know, but helps make my point) and if the 24" and 20" long sides are parallel and centered relative to each other (i.e. a line drawn from the center of one line and perpendicular to the same line will intersect the other line at it's center), the diagonal measurements will be identical, but obviously you don't have a rectangle (4 90 degree corners).

My guess is that your square is slightly off (say 1/32" in 24") and matches two of the angles in your panel. When you measure the other angles the amount of error that shows will be double the actual error so it will look like 1/16" in 24" or equivalent to the 1/8" in 4 ft that you mentioned.

By the way, the (a2 + b2) = c2 only works if one of the angles is a 90 degree angle.

Brian Walter

[Kevin Godshall]

Sounds to me like your framing square is off ... which would explain why you've got two corners that it fits nicely into and two that has gaps ...

EDIT: The reason I made the above statement was that I thought it was impossible to get two square corners and two unsquare corners in a single piece, but it dawns on me there IS one way to get this situation. Illustrated below with an exaggerated drawing:

115927

Took me awhile to get this straight in my head, but this is exactly (altho exaggerated) to hwhat I am ending up with. Didn't think was possible either at first. But you got it.

[Kevin Godshall]

One thing I haven't checked and will as soon as I get back to it..... is that both of my short ends measure the same, showing that either my long rips are parallel or they are not. I am assuming right now, they are not and that is where my problem is coming from.

I'm leaving for a weekend away from home, and I will not be doing calculations while I'm gone. Thanks to all for input. Will look forward to seeing additional comments when I get back Monday. Thanks!!

[Lee Schierer]

One other possability no one has mentioned is that one or more of your sides may be curved. This would give the impression that the part is square while the square is saying it isn't. Check each side with a long straight edge to be sure.

I usually check two adjacent corners for square with my square and measure the diagonals. If those measurements check out I go on with the project.

With regard to Mr. Pythagoras, even a small error on measuring each of the sides gets considerably larger when you sum the two squares.

[Peter Quinn]

Not to get away from the math or measuring, but how is the top constructed, and does it need to be square, or does it just need to be the same as what lies under it? If so, oversize and flush trim.

Another problem you may encounter in your search for square is that the edges of the rails of your square may not be parallel, and no amount of banging with a hammer is going to correct this. Are you making a plywood panel to fit into a rabbit ? If you are using the square to make one inside corner reference and one out side corner reference it may not work well.

I find measuring diagonals is QUITE precise if the instrument used to measure them is equally precise. A tape measure with a hook does not equal precise. Make a good pinch stick if you really need to get square. Basically two pieces of wood with a slot in each and a spline to keep them aligned, a bolt with a thumb screw to set the distance. You can make one with a notched hook on one end and a tapered point on the other for measuring outside corners, and one with a tapered point on each end for inside corners. It won't tell you what the actual distance of the hypotenuse is, but who really cares? It will tell you whether or not they are exactly the same, and if not by roughly how much. For squaring a carcass it can also tell you which leg is longer so you can adjust clamps to correct the problem.