Sine Bars

Part of Precision Measurement

Using the trigonometric sine relationship to set and measure precise angles using only length measurements.

Why This Matters

Angles are hard to measure accurately with direct methods. A protractor reads to 1 degree at best; an optical protractor to 5 arc minutes. But a sine bar can set an angle to 1 arc minute accuracy using nothing but gauge blocks and a surface plate — because it converts the angle measurement into a length measurement, and lengths can be measured very precisely.

This is an elegant example of a general principle: convert the hard problem into an easy one. We can’t easily measure angles to 0.01°, but we can easily measure lengths to 0.001 mm. Trigonometry connects the two.

In a precision workshop, sine bars are used whenever an angle must be set with accuracy better than what a protractor can provide. Setting up tapered bores for grinding, checking gear angles, verifying the taper on a morse taper arbor — all are sine bar territory.

The Sine Principle

A sine bar is a precision bar with two cylindrical rolls at each end, separated by an exactly known distance (typically 100 mm or 200 mm). When one roll rests on a surface plate and a stack of gauge blocks is placed under the other roll, the bar is inclined at an angle where:

sin(θ) = H / L

Where:

• θ is the angle
• H is the height of the gauge block stack under the elevated roll
• L is the distance between the roll centers

For a 100 mm sine bar:

• H = L × sin(θ)
• H = 100 × sin(θ)

Example calculations:

Angle (θ)	sin(θ)	Gauge block height for 100 mm bar
5°	0.08716	8.716 mm
15°	0.25882	25.882 mm
30°	0.50000	50.000 mm
45°	0.70711	70.711 mm
60°	0.86603	86.603 mm

Using a Sine Bar

Setting an angle:

1. Calculate H = L × sin(θ) for your desired angle
2. Build a gauge block stack equal to H from your set
3. Place the sine bar on the surface plate
4. Place the gauge block stack under one roll (the elevated end)
5. The bar is now inclined at exactly angle θ
6. Clamp your workpiece to the sine bar for grinding, inspection, or setup

Checking an unknown angle:

1. Place the tapered workpiece on the surface plate
2. Set a dial indicator to scan across the top face of the taper
3. Adjust gauge blocks until the dial indicator reads zero across the full length
4. Read the gauge block height H
5. Calculate θ = arcsin(H / L)

Compound Angles

For angles in two planes simultaneously, two sine bars are used — one for each plane. The second sine bar mounts on top of the first, rotated 90°. The geometry gets more complex but the principle is the same.

Making a Sine Bar

A sine bar is one of the simpler precision instruments to make, given:

• A lathe capable of precision cylindrical turning
• A surface grinder (or good hand-lapping skills)
• Hardened and ground cylindrical rod for the rolls

Components:

1. Body: Flat steel bar, typically 25 × 15 mm section, hardened
2. Rolls: Two cylinders of exactly equal diameter, hardened and ground
3. Roll mounting: Precision holes in the bar ends, or V-grooves, at exactly the design pitch

Critical requirements:

• The two rolls must be exactly equal in diameter (otherwise the effective pivot height differs)
• The roll center-to-center distance must be exactly the nominal value (100.000 mm, not 99.9 or 100.1)
• The top face of the bar must be flat and parallel to the line connecting roll centers

Making the rolls:

Turn two cylinders from tool steel to the same nominal diameter (say, 20 mm). Harden them. Grind to final diameter using the same setup without moving the grinder between pieces. This ensures equal diameter. Verify with a micrometer — they must agree within 0.002 mm.

Setting roll spacing:

Drill and ream two holes in the bar body at the design pitch. The reaming must be done very accurately — a 0.05 mm error in roll spacing produces a 0.03% length error, which translates to 1 arc minute of angle error at 30°. Press-fit or carefully fit the rolls with a retaining compound.

Checking the finished bar:

1. Place on surface plate, both rolls resting on the plate
2. Scan top face with dial indicator — should read within 0.005 mm end-to-end
3. Place gauge blocks equal to H for a known angle
4. Measure the actual angle with an optical instrument if available
5. Document any systematic error

Accuracy Considerations

The accuracy of a sine bar depends on:

1. Roll diameter equality — unequal rolls shift the effective center height
2. Roll spacing accuracy — directly multiplies into angle error
3. Gauge block accuracy — the stack height determines the angle
4. Surface plate flatness — the base reference
5. Temperature — both the bar and gauge blocks expand with heat

At small angles (under 15°), sine bar accuracy is highest — the sine function is nearly linear and small errors in H produce small angle errors. At large angles (over 45°), the cosine of the angle is small, meaning errors in H produce disproportionately large angle errors. For angles over 60°, consider using the complementary angle (60° = measuring 30° on the inverted side).

Gauge Block Stacks for Sine Bar Use

Gauge blocks must be wrung (optically bonded) into a stack:

1. Clean all faces with clean cloth and solvent
2. Overlap the first two blocks slightly, then slide them into position
3. They will stick together — the combined height equals the sum of the parts
4. Add blocks one at a time, building the stack
5. The wring connection is surprisingly strong and extremely accurate

A good gauge block set allows any height to be built in 0.001 mm increments by combining blocks. For sine bar use, calculate H to 0.001 mm and build the exact stack from the available combination of blocks.

The sine bar, paired with gauge blocks and a surface plate, forms a measurement system that can set angles to arc minute accuracy without any optical instruments or specialized calibration equipment — just careful metalwork and good arithmetic.