Spur gear calculator

Module (m) determines the size of the gear teeth. It is the pitch diameter divided by the number of teeth, measured in millimetres. A larger module means larger teeth and, for the same number of teeth, a larger gear.

For example, a 20-tooth gear with module 2 has a pitch diameter of 40 mm. Increasing the module to 3 increases the pitch diameter to 60 mm.

Gears intended to mesh must have the same module and compatible tooth geometry, including the same pressure angle.

Number of teeth (z) must be a whole number. Gear A is the driving gear, and Gear B is the driven gear.

d = m × z

The pitch diameter is the diameter of an imaginary reference circle. When two standard external gears mesh at their nominal centre distance, their pitch circles touch and roll together without slipping.

The schematic shows these reference circles touching.

Circular pitch is the distance along the pitch circle from a point on one tooth to the same point on the next tooth. In this nominal zero-backlash model, the tooth thickness along the pitch circle equals half that circular pitch, π × m / 2. At module 2 that arc is 3.1416 mm. Tooth count and pressure angle do not change it. Tooth outline on screen compares this arc with the straight chord between the same two points.

da = m × (z + 2)

The outside diameter is measured across the tips of the teeth. For standard full-depth gears, each tooth extends 1 module above the pitch circle.

This is a nominal dimension; manufacturing tolerances are not included.

Tip arc thickness is the length of the arc along this circle between the two flanks. It depends on the pressure angle: a larger pressure angle makes that arc narrower. The worked comparison is under Pressure angle.

df = m × (z − 2.5)

The root diameter is measured across the bottoms of the spaces between teeth. It is a nominal standard dimension; the real curved shape at each root depends on the cutter. The standard dedendum is 1.25 modules, which sets this diameter and leaves nominal clearance under the tip of a mating tooth.

An involute flank exists only on or outside its base circle. If the root circle is inside the base circle, the drawing joins the flank to the root with a short straight radial line. If the root circle is on or outside the base circle, no radial line is needed.

This join is a simplified drawing construction, not a cutter-generated root fillet. Do not use these sharp-cornered roots as a manufacturing profile. They are not a manufacturing-validated root profile.

The pictures show the 20-tooth gear in the first table row, not the 41-tooth gear. At 41 teeth the radial join is only 0.0274 mm, too short to draw as its own band. At 42 teeth the root circle is outside the base circle, so the radial join disappears. At 50 teeth the root is further outside the base circle, and the join is absent there too.

db = m × z × cos(α)

Imagine a taut string unwinding from a circular drum. The end of the string traces an involute curve. The drum represents the base circle. The working flanks of the tooth follow this kind of curve. The string is a way to picture the curve; it is not how a gear tooth is cut.

A larger pressure angle produces a smaller base circle when the pitch diameter stays fixed. The comparison is under Pressure angle.

The pair view can show this circle with the Base circles overlay and does not draw the tooth. Gear A draws the involute flanks from this circle, with a simplified sharp root.

Pressure angle describes the direction in which one gear tooth pushes against its mating tooth. More precisely, it is the angle between the line of action (the direction of that force in the idealised gear model) and the tangent to the pitch circles where those circles touch. The usual modern value is 20°. 14.5° and 25° are also used. This calculator accepts pressure angles greater than 0° and less than 45°.

Increasing the pressure angle, with the module and the tooth count fixed, makes the base circle smaller. The involute flank shape changes, and the tip arc thickness changes. The pitch diameter, the outside diameter and the nominal pitch-circle arc thickness stay the same.

The table is a worked example, not the gear typed above. The pictures show the circles, the pitch point and the pressure angle. They are not a tooth profile. The involute flanks are drawn only in the Gear A and Tooth views.

The working sides (flanks) of the drawn teeth follow a calculated involute curve. The screen represents each smooth curve with many short straight segments. That is an approximation for display, not a cutter path. The involute flanks are mathematically derived, but the roots are simplified and are not a cutter-generated fillet or a manufacturing-validated profile. When that join appears is explained under Root diameter.

Tooth thickness can be measured two ways: along the circle as an arc, or straight between the same two points as a chord. The calculator reports arc thickness. An arc is slightly longer than its chord. The diagram exaggerates that small difference to make it visible. Do not read the diagram as a tooth.

The differences are small; figures are rounded for display and are not manufacturing tolerances. Chord thickness is not a calculator result.

a = m × (zA + zB) / 2

The centre distance is the distance between the two gear axes.

For standard external gears without profile shift, it is the sum of the pitch radii. This is why the pitch circles touch in the schematic. With module 2, a 20-tooth gear and a 40-tooth gear have a standard centre distance of 60 mm.

Modified centre distances and backlash allowances are not included.

Gear ratio = zB / zA

The gear ratio is the number of teeth on the driven gear divided by the number on the driving gear.

If Gear A has 20 teeth and turns at 1,000 rpm, while Gear B has 40 teeth, Gear B turns at 500 rpm in the opposite direction. This is a 2:1 speed reduction. The formula below gives the speed magnitude. The direction is separate: two externally meshing spur gears rotate in opposite directions.

If you enter a driving speed, the calculator uses:

nB = nA × zA / zB

Internal gears are not supported.

With very few teeth, the generating cutter can remove material near the bottom of a tooth. This is called undercutting. It can weaken the tooth and shorten the useful involute contact surface.

For standard full-depth, unshifted gears, this calculator uses the approximate check z_min = 2 / sin²(α).

At a pressure angle of 20°, the calculated threshold is about 17.10 teeth. Since tooth counts are whole numbers, 17 teeth or fewer trigger the warning when the other inputs are valid; 18 teeth do not under this rule alone.

For a flagged Gear A, the tool shows reference dimensions but does not draw an unsupported involute tooth profile.

This is a simplified screening rule, not a guarantee of manufacturability, tooth strength or safe meshing.