Sine Bar Calculator (Gauge Block Stack)
Gauge block stack for any sine bar angle, the angle from a stack, and taper conversions - with the accuracy the setup really has at that angle.
How to use this calculator
- 1Set the bar on a surface plate with one roll down and the gauge block stack under the other.
- 2Build the stack from the fewest blocks that will make the height, and wring them properly.
- 3Above 45 degrees, set the complement and stand the work on its other face - the sine bar is much more accurate there.
- 4Indicate along the top face to check the work. A sine bar sets an angle; the indicator is what measures against it.
- 5For taper work, remember that taper per foot is on diameter - the compound slide is set to half the included angle.
How the calculation works
Stack = bar length x sin(angle)
Angle = arcsin(stack / bar length)
Sensitivity: dh = L x cos(angle) x da
Taper per foot = 2 x tan(included/2) x 12
Tailstock offset = (taper per inch / 2) x length between centres- L
- Distance between the roll centres - 5 inch and 10 inch are the standard sizes, 100 and 200 mm in metric
- Stack
- Gauge blocks under one roll, with the other roll on the surface plate
- cos(angle)
- The sensitivity factor. It is 1 at zero degrees and 0.5 at 60, so the same stack error costs twice the angle at 60 as it does flat
- Taper per foot
- Measured on diameter, so the half angle uses half of it. This is where most taper arithmetic goes wrong
A 10 inch bar gives twice the stack of a 5 inch bar for the same angle, which is twice the resolution for a given gauge block error.
Angles in degrees, minutes and seconds are sixtieths, not hundredths. 30 degrees 30 minutes is 30.5 degrees, not 30.30.
Tailstock offset depends on the whole length between centres, not the length of the tapered portion - which is why the method does not repeat across different workpiece lengths.
Worked example
Thirty degrees on a five inch bar
- 1.Stack = 5 x sin(30°) = 5 x 0.5 = 2.5000 in exactly - the tidiest setup there is.
- 2.The sensitivity factor is cos(30°) = 0.866, so a tenth of a thou in the stack moves the angle about 4.76 arc seconds.
- 3.That is 1.15 times worse than the same error would be at zero degrees.
- 4.Still comfortably inside anything a shop will measure, so setting it directly is fine at this angle.
Result: 2.5000 in stack - about 4.8 arc seconds per tenth
A steep angle, and why to set the complement
- 1.75 degrees gives a stack of 5 x sin(75°) = 4.8296 in - a tall, precarious setup.
- 2.Worse, cos(75°) = 0.2588, so a tenth of a thou now moves the angle 15.9 arc seconds - over three times worse than at 30 degrees.
- 3.The complement is 15 degrees, a stack of 1.2941 in.
- 4.Set that instead and stand the work on its adjacent face: the same tenth then costs only 4.27 arc seconds. Same angle, a quarter of the error, and a stack that will not topple.
Result: 4.8296 in direct - or 1.2941 in on the complement, four times more accurate
An angle in degrees and minutes
- 1.12 degrees 30 minutes is 12.5 degrees - minutes are sixtieths, so 30 minutes is half a degree, not 0.30.
- 2.On a 10 inch bar the stack is 10 x sin(12.5°) = 2.16440 in.
- 3.The same angle on a 5 inch bar would be 1.08220 in, so the longer bar doubles the stack.
- 4.Doubling the stack halves the angular effect of any gauge block error, which is the whole argument for the 10 inch bar.
Result: 2.16440 in on a 10 inch bar - twice the resolution of a 5 inch
A Morse taper, three ways
- 1.0.600 in per foot is 0.05 in per inch, on diameter.
- 2.The half angle is arctan(0.05/2) = arctan(0.025) = 1.4321 degrees, so the included angle is 2.8642.
- 3.To turn it by tailstock offset over 8 in between centres: (0.05/2) x 8 = 0.2000 in.
- 4.To turn it on the compound, set 1.4321 degrees - the half angle, because the compound follows one side of the cone.
Result: 2.8642° included, 0.2000 in tailstock offset over 8 in
A right triangle you can wring together
A sine bar is two hardened rolls of equal diameter set a precise distance apart in a flat bar - five inches or ten in the inch world, a hundred or two hundred millimetres in metric. Put one roll on a surface plate and lift the other on a stack of gauge blocks, and the roll centres form the hypotenuse of a right triangle whose opposite side is the stack.
So the stack is simply the bar length times the sine of the angle wanted. Five inches at thirty degrees is exactly 2.5000, which is the setup every apprentice learns first because the arithmetic is so clean.
What makes it capable of arc seconds is not the trigonometry, which any protractor could match, but the fact that gauge blocks are among the most accurate artefacts in a workshop. A grade 0 block is good to a few millionths, and wrung stacks add up faithfully. The sine bar converts that linear precision into angular precision, and that conversion is where all the subtlety lives.
Why steep angles are the problem
The conversion is not equally good at all angles, and this is the part most references leave out.
Differentiate the relationship: h = L sin(a) gives dh = L cos(a) da. Read backwards, an error in the stack produces an angular error divided by L cos(a). At small angles the cosine is close to one and the bar is at its most sensitive - a small stack error barely moves the angle. As the bar stands up, the cosine falls, and the same stack error produces progressively more angular error.
At 30 degrees the cosine is 0.866, so things are 15% worse than flat. At 60 degrees it is 0.5, so exactly twice as bad. At 75 degrees it is 0.259, nearly four times as bad. And at 80 or above the bar has effectively stopped being a precision instrument, quite apart from being a tall stack of blocks that would rather fall over.
The remedy is standard toolroom practice: above 45 degrees, set the complement. To get 75 degrees, set 15 degrees and arrange the work so the angle is taken from the adjacent face. The stack is smaller, more stable, uses fewer blocks, and sits on the accurate part of the curve. It costs nothing but deciding which face goes down.
The stack itself, and how it goes wrong
Gauge blocks are wrung together - slid into contact so that they adhere through a combination of surface contact and molecular attraction. A properly wrung stack behaves as one block. A stack that has been placed rather than wrung has air films between the blocks and is worth nothing at this level.
Every joint also contributes a small error, and errors accumulate with the number of blocks rather than with the height. So the rule is to use as few blocks as will make the number: start from the largest that fits under the target and work down, taking the thin blocks last. Making 2.1644 from four blocks is better than making it from seven, even though both arrive at the same nominal height.
Temperature matters more than people expect at these tolerances. Steel expands about 6.5 millionths per inch per degree Fahrenheit, so a warm hand on a stack of blocks for a minute is a real and measurable change. Blocks are meant to be handled minimally and allowed to settle to the plate temperature.
Tapers, and the factor of two that catches everyone
Taper per foot is stated on diameter. That single convention is responsible for most taper arithmetic errors, because almost everything you then want to do with it involves one side of the cone rather than both.
The half angle - which is what a compound slide is set to, since the tool follows one flank - is the arctangent of half the taper per inch. For a Morse taper at 0.600 in/ft, that is arctan(0.025) = 1.4321 degrees, and the included angle is twice it at 2.8642. Setting the compound to the included angle produces a taper twice as steep as intended, and it is a mistake experienced people still make.
Tailstock offset has its own trap. Offsetting the tailstock swings the whole axis of the work, so the offset needed depends on the total length between centres, not on the length of the tapered portion. The same taper on a longer piece needs a larger offset. It is why the method suits one-off work and why a taper attachment, which sets an angle rather than an offset, repeats across different lengths and the offset method does not.
The standard machine tapers are worth knowing as a family. The shallow ones - Morse at about 0.600 in/ft, Brown and Sharpe at 0.500, Jarno at exactly 0.600 - are self holding: the angle is small enough that friction retains the tool without a drawbar. The 7:24 milling tapers, at 3.500 in/ft, are deliberately steep so they release cleanly and locate rather than grip, which is why they need a drawbar to hold them in.
What this assumes, and where it stops
Assumptions
- The sine bar rolls are equal in diameter and the length quoted is centre to centre between them.
- The surface plate is flat and the setup is clean - a chip under a roll is a larger error than anything else here.
- Gauge blocks are properly wrung and at the same temperature as the plate.
- Sensitivity figures use one tenth of a thousandth of an inch, or one micrometre on a metric bar, as the reference stack error.
- Taper per foot is stated on diameter, which is the universal convention.
Limitations
- It does not model gauge block grade, wringing error, or accumulated error from the number of blocks.
- Compound angle sine plates - which set two angles at once - need each axis solved separately and interact in ways this does not cover.
- Thermal effects are noted but not calculated.
- The tailstock offset figure assumes the work is turned between centres with the taper along its axis.
- It does not check whether a compound slide has the travel for the taper length involved.
Common questions
How do I calculate a sine bar gauge block stack?
Multiply the bar length by the sine of the angle. A 5 inch bar at 30 degrees needs 5 x 0.5 = 2.5000 in of blocks. A 10 inch bar at the same angle needs 5.0000 in - twice the stack, which is twice the resolution for the same gauge block error.
Why is a sine bar less accurate at steep angles?
Because the stack height changes by L x cos(angle) per unit of angle, so as the angle rises the cosine falls and the same stack error produces more angular error. At 60 degrees it is twice as bad as flat; at 75 degrees nearly four times. Above 45 degrees, set the complement and stand the work on its other face.
Should I use a 5 inch or a 10 inch sine bar?
A 10 inch bar gives twice the stack for the same angle, so a given gauge block error produces half the angular error. It is the more accurate instrument. The 5 inch is more common because it is easier to handle, needs a smaller block set, and is accurate enough for most shop work.
How do I convert taper per foot to an angle?
Taper per foot is on diameter, so halve it first. The half angle is arctan(taper per inch / 2) and the included angle is twice that. A 0.600 in/ft Morse taper is arctan(0.025) = 1.4321 degrees per side, 2.8642 degrees included.
How much do I offset the tailstock to turn a taper?
Half the taper per inch, times the total length between centres. For 0.600 in/ft on an 8 inch job that is (0.05/2) x 8 = 0.200 in. Note it depends on the whole length between centres, not the tapered part, so the same taper on a different length of stock needs a different offset.
Sources
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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