How Square is Your Square?
 
A common question that comes up in woodworking forums is "How do I know if my square is really square?". Without access to an equipped metrology lab, this question can be rather difficult to answer with great accuracy, but we can come close enough to know if we have a serious problem with a square.
 
The most common method for checking if a square is a true 90 degrees involves using, what I call, the Pencil-Line-And-Flip Method (discussed last below).
 
An alternate way that I am proposing (if anyone finds a reference elsewhere to this procedure I'd appreciate you letting me know) involves using a dial indicator and a miter gauge at your table saw.
 
Dial Indicator Method

Procedure: Tilt your miter gauge to approximately 5 degrees1 (this doesn't have to be exact - approximate) and place your square with the smaller edge against the miter gauge. Place a stationary dial indicator (a magnetic base helps here) to the left side and position the stylus against the edge of the square at the top (d1 in Figure 1). To make calculations easier you can zero your dial indicator face then push the square and miter gauge forward and take a reading at the bottom of the square (d2 in Figure 1). Calculate Delta1 by taking the absolute value of the difference between distance d2 and distance d1 (Equation 1). Now flip the square over and repeat the same process to obtain Delta2 with the dial indicator on the right sde. In order to find out if your square is out of square use Equation 2, where L is the length of your square. The answer given will be the amount in degrees that your square is off. Try to keep the dial indicator plunger as parallel to the table saw as possible to reduce error.

Use the online calculator here.

Caveats:
  • A caveat to this method is that the edge of the square must be thick enough to position a dial indicator stylus against.
  • There are more measurements that must be taken than the 'Pencil Line and Flip Method' discussed below.
 
Figure1: d1, d2, d3 and d4 are the distances measured between the dial indicator and the square (red lines denote the distance measured.). You can zero your dial indicator on the first reading to obtain a Delta (the difference between d1 and d2 or d3 and d4). The two circles to the left of the square and the two to the right are the relative positions of the dial indicator before and after pushing the square and miter gauge forward.
 
 
Equation1
 
 
Equation2
 

Example: In Figure 1 the 8" long square shown has an angle error of 0.5 degrees (it's 89.5 degrees - acute). Delta1 is 0.767" and Delta2 is 0.628". Equation 2 tells us that the angle error is 0.498 degrees. The calculation is off by 0.002 degrees (see reference 2 for reason).

Figure2: An isosceles trapezoid which represents the squares shown in Figure1 with the same 0.5 degree angle error.. The distance denoted by the blue line is the difference between d1 and d2 measured in Figure1. The distance denoted by the green is the difference between d3 and d4 from Figure1. See example 1.
How accurate is this method? This depends on the accuracy of your dial indicator and the length of your square. Most dial indicators are accurate to 0.001" which would equate to 0.003 degrees of sensitivity with an 8" square.

Pencil Line and Flip Method

Procedure: Place your square on a piece of paper that is on a flat surface. Put the smaller length of your square against a stationary straight edge and draw a line (red line in Figure 2). Flip the square over and draw a second line (blue). Any gap that results between the two lines is a result of your square's angle error. The angle error can be calculated using Equation 3 (L is the length of the square).

Caveats:
  • A minor caveat to this method is that it can be difficult to discern small gaps between the two pencil lines (especially with a thick pencil lead). The most you might be able to detect is a 0.010" difference which equates to a minimum detection of 0.036 degrees with an 8" square.
  • Another caveat is that the edge you place your square against must be perfectly flat, otherwise you will not get an accurate calculation of your square's angle error.
Equation 3:
 
Figure 2
 
Conclusion: Both methods above are great for determining the amount of angle error for a square. The dial indicator has the potential to be 10-times more accurate when calculating the square's angle error (0.003 degrees min angle error detection vs 0.036 degrees min angle error detection).
 
References:
1) The angle of your miter gauge must be greater than the angle error (difference from 90, i.e. 0.5 degrees angle error for a 89.5 degree square) of your square.
2) Too large of a miter gauge angle (>5 degrees) will cause excessive errors in the calculation because as the miter gauge angle increases, the relative square length L (i.e. distance traversed for the two dial indicator readings d1 and d2 or d3 and d4) becomes shorter.