Calculate the gauge block height to set a sine bar, sine plate or vise to an angle, with a 5 in bar as the default, or find the angle from a block stack.
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 calculates ideal height from calibrated roll-center distance and angle, entered in decimal degrees or degrees, minutes and seconds. It can suggest up to five distinct blocks from the listed Mitutoyo 81-piece inch or 87-piece metric inventory after rounding to 0.0001 in or 0.001 mm. The result reports that nominal rounding residual separately; a suitable combination is not guaranteed. The local first-order sensitivity uses a reference height error of ±0.001 in for inch display or ±0.02 mm for metric display. The inverse mode returns decimal degrees, DMS and the complement. These calculations exclude bar, block, contact, surface and temperature uncertainty.
Gauge Block Heights and Error Sensitivity
The first table lists rounded ideal stack heights for common angles on 5 in and 10 in sine bars. A 5 in bar at 30 degrees needs an ideal height of 2.5000 in. Use the calibrated roll-center distance for the actual bar.
| 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 the local first-order sensitivity to height error: ΔA ≈ ΔH / (L × cos(A)) radians. Sensitivity increases as the angle approaches 90 degrees. These values use a 5 in bar and a reference 0.001 in height error; a 10 in bar halves this particular height-error contribution. The estimate excludes bar-length and other uncertainty and does not guarantee a finite symmetric error interval.
| 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 with a nominal 3.4202 in stack gives A = asin(3.4202 / 10) ≈ 19.999991 degrees. This rounds to 20° 0′ 0″ in DMS at 0.1 arc-second precision; the rounded stack does not give exactly 20 degrees.
FAQ
Why are sine bars made in 5 in and 10 in lengths?
The calibrated dimension is the center distance between the two rolls. The ideal stack height is that distance times the sine of the angle. For the same small height error and angle, doubling the roll-center distance halves the first-order height-error contribution; other uncertainty and practical constraints still need assessment.
Why should a sine bar not be used above 45 degrees?
Height-error sensitivity grows with 1/cos(A), and setup stability also needs assessment at high angles. Results above 45 degrees therefore carry a caution. The calculator reports the geometric complement, 90 degrees minus the angle; selecting a different physical setup requires appropriate equipment and metrology judgment. Exactly 90 degrees is excluded because the sensitivity is singular.
How is the gauge block stack actually assembled?
The suggestion searches nominal combinations of up to five distinct blocks from one listed inventory; it does not add duplicate or unavailable sizes. The Mitutoyo 81-piece inch inventory contains 0.1001–0.1009 in by 0.0001 (9), 0.101–0.149 in by 0.001 (49), 0.05–0.95 in by 0.05 (19), and 1–4 in by 1 (4). The listed 87-piece metric inventory contains 1.001–1.009 mm by 0.001 (9), 1.01–1.49 mm by 0.01 (49), 0.5–9.5 mm by 0.5 (19), and 10–100 mm by 10 (10). Other sets can differ. The height is rounded to the selected inventory resolution and the residual is reported. A suggested nominal sum is not a guarantee of availability, calibration, setup suitability or minimum physical uncertainty.
