mm to Degree Calculator

Convert mm to degrees and degrees to mm for an offset over a run (mm per metre by default), an arc on a circle, wheel toe or a control surface.

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Calculator Functions

Function	What it does	What it requires
mm to degrees (offset over a run)	Converts an offset in mm to the angle it makes over a run	Requires the offset and the run, which starts at 1,000 mm
Degrees to mm (offset over a run)	Converts an angle to the offset it produces over a run	Requires the run and the angle in degrees
Run from offset and angle	Finds the run over which an offset makes a given angle	Requires the offset and the angle in degrees
Arc on circle — angle	Finds the angle an arc length covers on a circle	Requires the arc length and the radius
Arc on circle — arc length	Finds the length along a circle for a given angle	Requires the radius and the arc angle in degrees
Arc on circle — radius	Finds the radius at which an arc length covers a given angle	Requires the arc length and the arc angle in degrees
Wheel toe — angle	Converts a wheel toe measurement in mm to a total toe angle and a per-wheel angle	Requires the toe amount, the measurement diameter, total or one-wheel toe and the direction
Wheel toe — toe amount	Converts a total toe angle to the toe amount in mm	Requires the measurement diameter, the total toe angle, total or one-wheel toe and the direction
Wheel toe — measurement diameter	Finds the diameter at which a toe amount equals a given total toe angle	Requires the toe amount, the total toe angle, total or one-wheel toe and the direction
Control surface — angle	Converts the deflection of a hinged surface to its angle	Requires the deflection and the surface width from hinge to measuring point
Control surface — deflection	Converts a hinged surface angle to its deflection	Requires the surface width and the surface angle in degrees
Control surface — surface width	Finds the surface width at which a deflection equals a given angle	Requires the deflection and the surface angle in degrees

Lengths go in as mm, cm, inches or meters, except wheel toe, which uses mm or inches. A length result can be shown in mm, cm, inches or meters. Every result gives the angle in decimal degrees and in degrees and minutes, and the offset functions add the slope in percent and in mm per meter.

mm to Degree Formulas

A length in mm has no angle of its own. It becomes an angle only against a second length: a run, a radius, a wheel diameter or a surface width. The four groups of functions use different geometry, so the same two numbers give different angles in each.

mm to degrees (offset over a run)

The main formula applies directly. An offset of 10 mm over a 1,000 mm run is atan(10 ÷ 1,000) × 180 ÷ π = 0.5729°, which is 0° 34.38′ in degrees and minutes. The table uses a 1,000 mm run, so each offset is also the slope in mm per meter.

Offset over 1,000 mm (mm)	Angle (degrees)
1	0.0573
2	0.1146
5	0.2865
10	0.5729
20	1.1458
25	1.4321
50	2.8624
100	5.7106
200	11.3099

Degrees to mm (offset over a run)

One degree over a 1,000 mm run is 1,000 × tan(1°) = 17.4551 mm. The offset scales with the run, so the same angle over 100 mm is one tenth as much.

Angle (degrees)	Offset over 1,000 mm (mm)	Offset over 100 mm (mm)
0.1	1.7453	0.1745
0.25	4.3634	0.4363
0.5	8.7269	0.8727
1	17.4551	1.7455
2	34.9208	3.4921
5	87.4887	8.7489
10	176.3270	17.6327
15	267.9492	26.7949
30	577.3503	57.7350
45	1000.0000	100.0000

Run from offset and angle

An offset of 50 mm at 5° needs a run of 50 ÷ tan(5°) = 571.50 mm. An angle of zero has no answer.

Arc on circle — angle

s is the length measured along the circle and r is the radius. A 50 mm arc on a 100 mm radius covers 50 ÷ 100 × 180 ÷ π = 28.6479°.

Arc on circle — arc length

An angle of 30° on a 100 mm radius is an arc of 100 × 30 × π ÷ 180 = 52.3599 mm.

Arc on circle — radius

A 50 mm arc that covers 30° lies on a radius of 50 ÷ 0.523599 = 95.493 mm.

Wheel toe — angle

t is the toe amount, d is the diameter at which it is measured, and k is 2 for total toe across both wheels or 1 for one wheel. θ is the total toe angle, and each wheel sits at half of it. A total toe of 5 mm measured on a 381 mm (15 in) diameter is 2 × asin(5 ÷ 762) = 0.7519°, or 0.3760° per wheel.

Wheel toe — toe amount

A total toe angle of 0.5° on a 381 mm diameter is a total toe of 2 × 381 × sin(0.25°) = 3.3248 mm.

Wheel toe — measurement diameter

A total toe of 3 mm equals 0.4° when it is measured on a diameter of 3 ÷ (2 × sin(0.2°)) = 429.72 mm.

Control surface — angle

y is the deflection at a right angle to the neutral surface and w is the width from the hinge to the measuring point. A 40 mm deflection on a 50 mm wide surface is asin(40 ÷ 50) = 53.1301°. The deflection cannot be larger than the width.

Control surface — deflection

A 50 mm wide surface at 30° deflects 50 × sin(30°) = 25 mm.

Control surface — surface width

A 20 mm deflection at 15° belongs to a surface 20 ÷ sin(15°) = 77.274 mm wide.

Limits of This Estimate

• The offset functions assume the offset is at a right angle to the run, and the angle stays between −90° and 90°.
• Arc length is measured along the curve. A straight chord between the same two points is shorter and gives a different angle.
• The wheel toe functions assume both wheels carry equal toe and that the diameter is the one at the measuring points, such as the rim edge or the tire tread. Toe quoted at a different diameter converts to a different angle.

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