The whole of it is one triangle
A sine bar is two rolls of equal diameter held a known distance apart. Stand one roll on a surface plate, put a stack of blocks under the other, and the bar becomes the hypotenuse of a right triangle whose opposite side is the stack. Opposite over hypotenuse is the sine, so the stack is the centre distance multiplied by the sine of the angle you want. Nothing else in the arrangement matters — not the roll diameter, not the length of the bar past the rolls, not the thickness of the bar.
On the default setting that is 5 in times sin 30, which is 2.5000 in exactly. The exactness is a coincidence of thirty degrees and it is worth noticing how quickly it stops being convenient: 12 degrees on the same bar wants 1.0396 in, and 12 degrees 30 minutes wants 1.0821.
Why the tangent is such a persistent mistake
Under five degrees the sine and the tangent of an angle agree to about four decimal places, so a setup built on the wrong function comes out right and nobody learns anything. At 10 degrees on a 5 in bar the two differ by 13.4 thou. At 30 degrees they differ by 386.8 thou, and the part comes off with 35.26 degrees on it instead of 30. At 45 degrees the tangent asks for a 5.0000 in stack under a 5 in bar, which would stand the thing vertical, and only then does the arrangement look obviously wrong.
The page prints both numbers side by side for whatever angle you entered, and the difference between them is the honest measure of how much the error would have cost on that particular job.
The residual is the interesting number
Ask for 2.5000 in from the default block list and it comes out of one block. Ask for 1.0821 in — which is 12 degrees 30 minutes on the same bar — and the greedy build takes 1, then finds nothing that fits until 0.0821 is left, and it cannot close it, because the smallest listed block is 0.1001. Edit the list to include the small end of a real set and the residual closes; leave it as it is and the page tells you plainly that it did not.
That residual is not a rounding artefact. It is the actual angle error you would set if you built the stack from those blocks and stopped, and the page converts it into arcseconds so it can be compared against the tolerance on the print rather than admired as a decimal.
What a tenth is worth changes with the angle
The rate at which the angle moves for a given stack error is one over the centre distance times the cosine. On a 5 in bar at 30 degrees, a tenth of stack error is 4.76 arcseconds. At 60 degrees the same tenth is 8.25 arcseconds, and at 80 it is 23.7. This is why steep angles are set on a compound sine plate or an angle block rather than on a bar, and it is also why a nominally 5 in bar that is actually 4.9990 matters more at the top of the range than the bottom.
The bar length error works the other way round. Setting the nominal stack on a bar that is a tenth short shifts the angle by 2.38 arcseconds at 30 degrees and by 7.15 at 60. Both effects are computed for the angle in the form rather than quoted as a rule, because they are ratios and the ratios move.
What the page deliberately does not do
It does not know your bar. The centre distance in the box is the number the arithmetic uses, and if the bar has been dropped, reground or simply never checked, the arithmetic is faithful to a wrong input. It does not know whether the plate is flat, whether the blocks are clean, whether the stack is wrung or merely stacked, or whether the work sitting on the bar is pulling it off the plate. It also has no opinion on whether the resulting angle is inside the tolerance on the drawing. All of that is bench work and inspection, and it is where the real error in an angular setup nearly always lives.
Questions people ask
What is the formula for a sine bar stack height?
Stack height equals the centre distance between the rolls multiplied by the sine of the angle. On a 5 in bar at 30 degrees that is 5 times 0.5, or 2.5000 in. The roll diameter, the overall length of the bar and the thickness of the bar do not enter into it — only the distance between the roll centres does.
Why not use the tangent for a sine bar?
Because the stack is the side opposite the angle and the bar is the hypotenuse, which makes it a sine. The tangent would be right if the bar lay along the plate, and it does not. Under about five degrees the two agree closely enough to hide the mistake; at 30 degrees on a 5 in bar the tangent asks for 2.8868 in instead of 2.5000, and the part comes out at 35.26 degrees.
How accurate does the gauge block stack have to be?
It depends on the angle and the bar. On a 5 in bar a tenth of an inch error in the stack moves the angle by 4.76 arcseconds at 30 degrees, 8.25 at 60 and 23.7 at 80. The page works the figure out for the angle you entered so it can be compared with the angular tolerance on the print rather than guessed at.
Why can the calculator not always build the stack I asked for?
Because it only uses the block sizes you list, each one once, up to the limit you set. If the sizes in your list cannot sum to the target, it stops and prints how far short it is and what that residual is worth in arcseconds. Adding the small end of your set to the list usually closes it. The picking is greedy rather than exhaustive, so a different combination of the same blocks can sometimes do better.
Does the sine bar formula work for a sine plate?
Yes, about its own hinge to roll centre distance, which is not usually the same number as a sine bar and is stamped on the plate. A compound sine plate is two of them at right angles and each stack is worked from its own centre distance and its own angle. What the page cannot tell you is which reference face the angle on your print is measured from, and that is the more common source of a wrong setup than the arithmetic is.