Sine Bar Calculator
The gauge block stack for any angle — and the angle you actually get from a stack you already have. Drawn to scale, with the achievable resolution shown, because the useful question on the surface plate is usually "how close can I actually get with the blocks in the drawer?"
How a sine bar works, and where its accuracy comes from
A sine bar is two precision rollers set a known centre distance apart on a hardened bar. Raise one roller on a stack of gauge blocks and the bar tilts to a known angle — the geometry is a right triangle where the bar is the hypotenuse and the stack is the opposite side. So H = L × sin(θ), and going the other way, θ = arcsin(H ÷ L). That is the entire principle, and it is why a sine bar is genuinely accurate: it converts an angle into a length, and we can measure length far better than we can measure angle directly.
The interesting consequence of the sine relationship is that resolution is not constant. Because sine flattens as it approaches 90°, the same 1 µm error in your stack produces a very different angular error at 5° than at 60°. Near zero, the stack height changes quickly with angle, so you can set angles finely. Near 45° it is still comfortable. Past about 60° the sine curve is flattening and the same block error costs you noticeably more angle — which is exactly why the conventional guidance is to avoid using a sine bar much beyond 45–60° and to use an angle block or a sine plate stacked differently instead. This calculator shows the resolution at your specific angle rather than leaving you to guess.
On standard lengths: 100 mm and 200 mm are the common metric centre distances; 5 inch (127 mm) and 10 inch (254 mm) are the imperial equivalents you will still find in most inspection rooms. Longer is generally better — a 200 mm bar needs twice the stack of a 100 mm bar for the same angle, and because the error is a proportion of the stack, the longer bar gives you a finer angular resolution for the same block set.
The practical limits are the boring ones. Gauge blocks come in finite steps, so the achievable angle is quantised — you usually cannot hit an arbitrary angle exactly, and this tool tells you how close you can get and what the residual error is. Blocks must be wrung properly, not just piled up. And everything here assumes 20°C: steel moves roughly 11 µm per metre per °C, so on a warm afternoon a tall stack is not the height you think it is. At these resolutions, temperature is not a footnote.
Sine bar — FAQ
How do you calculate sine bar height?
Height = sine bar length x sin(angle). For a 100 mm sine bar at 30 degrees: 100 x sin(30) = 100 x 0.5 = 50.000 mm of gauge blocks.
How do you find the angle from a sine bar stack?
Angle = arcsin(stack height / sine bar length). A 50 mm stack on a 100 mm bar gives arcsin(0.5) = 30 degrees.
What length sine bars are standard?
100 mm and 200 mm are the common metric centre distances; 5 inch (127 mm) and 10 inch (254 mm) are the usual imperial ones. A longer bar needs a taller stack for the same angle, which gives finer angular resolution for the same gauge block set.
Why should you not use a sine bar above 60 degrees?
Because sine flattens as it approaches 90 degrees. The same small error in the gauge block stack produces a progressively larger angular error at high angles, so accuracy degrades. Above roughly 45 to 60 degrees, an angle block or a differently arranged sine plate is the better tool.
Does temperature affect a sine bar setup?
Yes. Steel expands roughly 11 micrometres per metre per degree C. At the resolutions a sine bar reaches, a tall gauge block stack measured away from 20 degrees C is not the height you assume it is. Inspection at a controlled temperature matters.
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