How a sine bar turns an angle into a height
A sine bar is a hardened, ground bar with two precision rolls of equal diameter set a known distance apart — almost always 5 or 10 inches between roll centres. Rest one roll on a surface plate, lift the other on a stack of gauge blocks, and the bar sits at an angle.
The geometry is a right triangle. The roll centre distance is the hypotenuse, the stack height is the opposite side, and the angle between the bar and the plate is what you are setting. So sin θ = H / L, and rearranged, H = L × sin θ. That is the whole method, and it is exact — no small-angle approximation, no correction factor.
The reason it is trusted for precision work is that it converts an angle, which is hard to measure directly to any accuracy, into a length, which gauge blocks measure to millionths. The accuracy of the setup is the accuracy of the blocks and the bar, and both are very good.
Why bars are 5 and 10 inches, and why it matters which you use
Five inches is convenient arithmetic. At 30 degrees, sine is exactly 0.5, so the stack is exactly 2.5000 — half the bar. Ten-inch bars double every number, which makes mental checks easy in the other direction.
The longer bar is the more accurate one, and the reason is worth understanding. A given error in the stack produces an angular error inversely proportional to the bar length, so a 0.0001 inch error on a 10 inch bar causes half the angular error it would on a 5 inch bar. If the setup is critical and the part fits, use the longer bar.
What does not change is the roll centre distance you must use in the arithmetic. It is not the overall length of the bar and it is not the distance between the roll edges — it is centre to centre, and it is the number stamped on the tool.
Worked example: setting 12° 30' on a 5 inch bar
- Convert to decimal degrees. 30 arc minutes is 30 ÷ 60 = 0.5 degrees, so the angle is 12.5°.
- Take the sine. sin 12.5° = 0.216440.
- Multiply by the bar length. 5 × 0.216440 = 1.08220 inch.
- Build the stack. From an 81-piece set, working from the smallest digit up: 0.1002 clears the ten-thousandths, 0.1320 clears the thousandths, 0.8500 clears the remainder. Three blocks, wrung together.
Check the direction of the error while you are there. From the table on this page, one extra arc minute at 12° adds about 0.00142 inch to the stack on a 5 inch bar. So a stack that is a tenth of a thou high — 0.0001 — is worth roughly four arc seconds. That is the resolution the method actually delivers, and it is why sine bars are still standard equipment.
Where the accuracy goes, and the 45 degree ceiling
Sine bars get less accurate as the angle rises, and the reason is the shape of the sine curve. Near zero degrees, sine changes almost linearly with angle, so a small stack error is a small angle error. Near 90 degrees the curve flattens out completely: enormous changes in angle produce almost no change in height, so the arithmetic becomes ill-conditioned in reverse — a tiny stack error implies a large angular one.
The practical rule is that a sine bar is reliable to about 45 degrees. Above that, use a sine plate, an angle block, or set the complement of the angle from the other face. Setting 70 degrees is usually done as 20 degrees from the perpendicular rather than 70 from the plate.
The other accuracy limit is the stack itself. Each wrung joint between blocks contributes a small uncertainty, so a stack of six blocks is measurably less trustworthy than a stack of three. The digit method used here minimises block count deliberately, and it is why the classic sequence — clear the ten-thousandths first, then the thousandths, then the coarse blocks — is worth following rather than picking blocks by eye.
Stack heights for a 5 inch sine bar
| Angle | Stack (in) | Per arc minute (in) |
|---|---|---|
| 1° | 0.0873 | 0.00145 |
| 5° | 0.4358 | 0.00145 |
| 10° | 0.8682 | 0.00143 |
| 15° | 1.2941 | 0.00140 |
| 20° | 1.7101 | 0.00137 |
| 30° | 2.5000 | 0.00126 |
| 45° | 3.5355 | 0.00103 |
| 60° | 4.3301 | 0.00073 |
Notice the last column shrinking. That falling sensitivity is exactly why the method loses precision at high angles.
Wring the blocks, do not stack them
Gauge blocks are meant to be wrung — slid together under light pressure until they adhere, with no measurable film between them. Blocks simply piled up trap air and contamination, and the stack reads high by an unpredictable amount. Wipe each face, wring each joint, and build the stack in the order the calculation gives so the smallest blocks are handled least.
Setting up accurately
- Use the roll centre distance. Not the bar's overall length, and not the outside dimension across the rolls.
- Clean the surface plate and both rolls. A chip under a roll is a direct angular error, and it is the most common cause of a setup that will not repeat.
- Check with an indicator. Sweep the bar's top face and confirm it reads flat along its length before trusting the angle.
- Keep the block count low. Each wrung joint adds uncertainty. Three blocks is comfortable; six is worth re-planning.
- Above 45 degrees, set the complement. Work from the perpendicular instead, or move to a sine plate.
Related setups and measurements
A sine bar sets an angle; several neighbouring jobs need the angle worked out first. Turning a taper on a lathe is the common one — the lathe taper calculator converts a large and small diameter into the included angle and the compound setting, and that angle is what you would then set on a sine bar to inspect the finished part.
For laying out holes on a rotary table or a mill without a DRO's bolt-circle function, the bolt circle calculator gives the coordinates directly. And where the setup is about fit rather than angle, the hole and shaft fit calculator covers the tolerance side of the same inspection work.
Terms used here
- Roll centre distance
- The distance between the centres of a sine bar's two rolls, stamped on the tool. The L in the formula.
- Wringing
- Sliding two gauge blocks together under light pressure so they adhere with no measurable gap. The only correct way to build a stack.
- Arc minute
- One sixtieth of a degree. An arc second is one sixtieth of that, and is roughly the resolution a 5 inch sine bar and good blocks can hold.
- Sine plate
- A sine bar built into a work-holding fixture, often with two axes, used where the part must be clamped as well as angled.