Calculate the exact gauge block stack height needed to set a sine bar to any angle, or find the angle produced by a known block height and bar length.
Sine Bar Formula
- H is the gauge block height, the total height of the block stack placed under the raised roll (in or mm)
- L is the sine bar length, measured center to center between the two rolls (in or mm)
- A is the angle the sine bar is set to (degrees)
The sine bar and the surface plate form a right triangle. The bar itself is the hypotenuse with a known length L, and the gauge block stack is the side opposite the angle. Because sin(A) equals opposite over hypotenuse, the stack height needed for any angle is simply L times the sine of that angle.
The first mode of the calculator solves the setup problem: enter the bar length and the target angle, in decimal degrees or in degrees, minutes, and seconds, and it returns the exact stack height. It also suggests a specific combination of blocks from a standard 81-piece inch set or 87-piece metric set, and reports how many arc-seconds the angle would shift if the stack height were off by 0.001 in (0.02 mm). The second mode solves the inspection problem in reverse: enter the bar length and a known stack height and it returns the angle in decimal degrees and DMS along with its complement.
Gauge Block Heights and Error Sensitivity
The first table lists the stack heights for common angles on the two standard imperial sine bars. Note that a 5 in bar at 30 degrees needs exactly 2.5000 in of blocks, which is why 5 in and 10 in became the standard lengths: the math stays clean.
| Angle | sin(A) | Stack, 5 in bar | Stack, 10 in bar |
|---|---|---|---|
| 5° | 0.08716 | 0.4358 in | 0.8716 in |
| 10° | 0.17365 | 0.8682 in | 1.7365 in |
| 15° | 0.25882 | 1.2941 in | 2.5882 in |
| 20° | 0.34202 | 1.7101 in | 3.4202 in |
| 25° | 0.42262 | 2.1131 in | 4.2262 in |
| 30° | 0.50000 | 2.5000 in | 5.0000 in |
| 35° | 0.57358 | 2.8679 in | 5.7358 in |
| 40° | 0.64279 | 3.2139 in | 6.4279 in |
| 45° | 0.70711 | 3.5355 in | 7.0711 in |
The second table shows why machinists avoid sine bars above 45 degrees. The angle error caused by a fixed stack height error grows with 1/cos(A), so the same 0.001 in mistake that costs about 41 arc-seconds at 5 degrees costs nearly six times that at 80 degrees. Values below are for a 5 in bar with a 0.001 in height error; a 10 in bar halves each value, which is the main reason to buy the longer bar.
| Set angle | Angle error per 0.001 in of stack error | In arc-minutes |
|---|---|---|
| 5° | 41.4 arc-sec | 0.7′ |
| 15° | 42.7 arc-sec | 0.7′ |
| 30° | 47.6 arc-sec | 0.8′ |
| 45° | 58.3 arc-sec | 1.0′ |
| 60° | 82.5 arc-sec | 1.4′ |
| 70° | 120.6 arc-sec | 2.0′ |
| 80° | 237.6 arc-sec | 4.0′ |
Example Problems
Example 1: Gauge block height from angle. You need to set a 5 in sine bar to 30 degrees. Multiply the bar length by the sine of the angle: H = 5 × sin(30°) = 5 × 0.5 = 2.5000 in. A clean two-block stack from an 81-piece set is 0.500 + 2.000.
Example 2: Angle from gauge block height. A 10 in sine bar is resting on a 3.4202 in stack. Divide the height by the bar length and take the inverse sine: A = asin(3.4202 / 10) = asin(0.34202) = 20.000 degrees, or 20° 0′ 0″ in DMS.
FAQ
Why are sine bars made in 5 in and 10 in lengths?
The calibrated dimension is the center distance between the two rolls, and making it a round number keeps the math simple: on a 5 in bar the stack height is just 5 times the sine of the angle. The 10 in bar has a second advantage beyond easy math: because angle error scales with stack error divided by bar length, a 10 in bar cuts the effect of any block or surface error in half compared with a 5 in bar.
Why should a sine bar not be used above 45 degrees?
The error in the set angle grows in proportion to 1/cos(A), so it climbs steeply as the angle rises, and the tall block stack itself becomes less stable. Above 45 degrees the standard practice is to set the complement of the angle (90 degrees minus the target) and reference the workpiece against a right-angle plate, which keeps the bar in its accurate low-angle range.
How is the gauge block stack actually assembled?
Blocks are chosen to eliminate the smallest decimal place first, using the fewest blocks possible, then wrung together by sliding the lapped faces so no air gap remains. A standard imperial 81-piece set contains nine tenth-thousandth blocks (0.1001 to 0.1009 in), forty-nine thousandth blocks (0.101 to 0.149 in), nineteen fifty-thousandth blocks (0.050 to 0.950 in), and four whole-inch blocks (1 to 4 in), which is why almost any height to 0.0001 in can be built from about four blocks. The calculator applies this same elimination method when it suggests a stack.