How do you calculate spur gears?

Start from the shared module m and each tooth count N: pitch diameter d = m × N, outside diameter = d + 2m, root diameter = d − 2.5m, circular pitch = π × m, and center distance C = m × (N1 + N2) ÷ 2. The gear ratio is N2 ÷ N1, output speed is input speed ÷ ratio, and output torque is input torque × ratio before friction losses.

What is the gear module?

The module (m) is the ratio of pitch circle diameter to number of teeth (m = d / N), in millimetres. Both gears in a meshing pair must have the same module. Preferred values in ISO 54 include 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 and 10 mm.

How do I convert diametral pitch to module?

Module m = 25.4 ÷ P, where P is diametral pitch in teeth per inch. A 10 DP gear is 2.54 module, 12 DP is 2.117 and 24 DP is 1.058, so inch and metric gears of “similar size” are usually not interchangeable. In inch gears the pitch diameter is d = N ÷ P, outside diameter (N + 2) ÷ P and center distance (N1 + N2) ÷ (2P).

What is the contact ratio and why does it matter?

The contact ratio is the average number of tooth pairs in contact: ε = [√(ra1² − rb1²) + √(ra2² − rb2²) − a·sin αw] ÷ (π m cos α). It must exceed 1 for continuous contact; a value of at least 1.2 is a common minimum and around 1.4 to 1.8 is typical for standard 20° spur gears.

What pressure angle should I use?

20 degrees is the modern standard for most spur gears and provides stronger teeth with lower sliding friction. 14.5 degrees is an older standard still used in replacement parts, and 25 degrees gives stronger teeth at the cost of a lower contact ratio. Both gears must use the same pressure angle.

What is profile shift on a spur gear?

Profile shift moves the cutting rack outward (positive x) or inward (negative x) by x × m. A positive shift thickens the tooth root, avoids undercut on small pinions and increases the center distance. For shifted gears the working pressure angle comes from inv αw = 2 tan α (x1 + x2) ÷ (N1 + N2) + inv α, and center distance a = (N1 + N2) m cos α ÷ (2 cos αw).

What is the minimum number of teeth to avoid undercut?

For a standard full-depth rack the limit is 2 ÷ sin²α: about 17.1 teeth at 20°, 32 at 14.5° and 11.2 at 25°. With profile shift the minimum shift is x = 1 − N·sin²α ÷ 2, so a 12-tooth 20° pinion needs x ≥ 0.30.

What is the gear center distance formula using module and teeth?

Center distance C = m × (N1 + N2) / 2 for unshifted gears. It equals half the sum of the two pitch diameters; for module 2 with 20 and 40 teeth, C = 2 × 60 / 2 = 60 mm between shaft centerlines.

How do I calculate spur gear module from center distance?

For an unshifted external gear pair, module m = 2C / (N1 + N2). If that gives a non-standard module, keep a standard module and use profile shift to make up the difference: this calculator can solve the required shift for a target center distance.

How do I calculate the total ratio of a two-stage spur gear train?

Multiply the stage ratios: total ratio = (N2 ÷ N1) × (N4 ÷ N3). For example, 2:1 followed by 3:1 gives 6:1. This calculator solves one meshing pair at a time, so run each stage separately.

Does this tool handle helical or bevel gears?

No — this tool is specific to external involute spur gears (parallel axes, straight teeth). Helical, bevel and internal gears use different geometry, and tooth strength ratings need AGMA 2101 or ISO 6336.

Spur Gear Calculator

Solve a spur-gear pair from teeth and module or diametral pitch, with profile shift, center distance, ratio, output speed and torque, contact ratio and every diameter.

Looking for bicycle gearing (gear inches, gain ratio, speed at cadence)? Use the bicycle gear calculator.

Gear Pair Inputs

Shared Parameters

Gear system

Module (m) mm

Pressure Angle

Target Center Distance (mm) (optional) Meet the target by

Driver (Pinion) — Gear 1

Teeth N1

Profile shift x1

Speed (rpm) (opt.)

Torque (N·m) (opt.)

Driven — Gear 2

Teeth N2

Profile shift x2

Gear Pair Summary

Gear Ratio

—

Center Distance

—

Contact Ratio

—

Working Pressure Angle

—

Driven Speed

—

Driven Torque (ideal)

—

Pitch-line Velocity

—

Tangential Tooth Force

—

Gear Geometry (mm)

Parameter	Driver (Gear 1)	Driven (Gear 2)
Number of Teeth	—	—
Profile Shift x	—	—
Pitch Diameter	—	—
Base Circle Diameter	—	—
Working Pitch Diameter	—	—
Addendum	—	—
Dedendum	—	—
Outside (Tip) Diameter	—	—
Root Diameter	—	—
Whole Depth	—	—
Circular Pitch	—	—
Tooth Thickness at Tip	—	—
Min. Shift to Avoid Undercut	—	—

Formula Reference

Pitch diameter d = m × N = N ÷ P Module m = 25.4 ÷ P Base circle d_b = d × cos α inv α = tan α − α inv α_w = 2 tan α (x1+x2)/(N1+N2) + inv α Center dist. a = (N1+N2) m cos α / (2 cos α_w) y = a/m − (N1+N2)/2 Tip dia. d_a1 = d1 + 2m(1 + y − x2) Root dia. d_f = d − 2.5m + 2xm Whole depth h = m(2.25 + y − x1 − x2) Circular pitch p = π × m Undercut x_min = 1 − N sin²α / 2 Gear ratio i = N2 / N1 T2 = T1 × i, n2 = n1 / i (no losses) Contact ratio ε = [√(r_a1²−r_b1²) + √(r_a2²−r_b2²) − a sin α_w] / (π m cos α)

Geometry follows the involute relations in ISO 21771 and KHK's gear technical reference for standard full-depth teeth (addendum coefficient 1, clearance 0.25). With profile shift, the tip is shortened by (x1 + x2 − y)m so the standard bottom clearance is kept. Torque and force ignore friction; a single spur mesh typically loses about 1–2%.

Quick answer

To calculate spur gears, multiply module by tooth count for each pitch diameter (d = m × N), add 2m for the outside diameter, subtract 2.5m for the root diameter, and set center distance C = m × (N1 + N2) ÷ 2. In inch gears, d = N ÷ P for diametral pitch P. Gear ratio is driven teeth ÷ driver teeth.

Example: Module 2 mm with 20 and 40 teeth: pitch diameters 40 mm and 80 mm, outside diameters 44 mm and 84 mm, center distance 60 mm, ratio 2:1, so 1,500 rpm in gives 750 rpm out and 10 N·m in gives 20 N·m out (before losses).

Enter teeth and module or diametral pitch, or a target center distance, to solve the full pair.

Summary

This calculator covers an external involute spur-gear pair with standard full-depth teeth (addendum 1m, dedendum 1.25m) in metric module or inch diametral pitch. It computes ratio, output speed and torque, pitch-line velocity, center distance, working pressure angle, contact ratio, pitch/base/working/outside/root diameters, tip thickness and undercut limits, with optional profile shift. A target shaft center distance can be met either by solving the module or, keeping a standard module, by solving the profile shift. Strength rating, backlash and tolerances still require AGMA 2101 / ISO 6336 and the applicable accuracy standard.

How it works

Pitch diameter d = m × N (metric) or N ÷ P (inch, with m = 25.4 ÷ P).
With profile shift x1 + x2, the working pressure angle αw solves inv αw = 2 tan α (x1 + x2) ÷ (N1 + N2) + inv α, where inv α = tan α − α.
Center distance a = (N1 + N2) m cos α ÷ (2 cos αw); with no shift this is m (N1 + N2) ÷ 2.
Tip diameters use addendum (1 + y − x_other) m so the standard 0.25m clearance is kept; root diameter = d − 2.5m + 2xm.
Contact ratio = path of contact ÷ base pitch; undercut and tip thickness are checked for each gear.

Use cases

Size a gear pair for a gearbox or power transmission system.
Fit a gear pair to an existing shaft spacing with profile shift instead of a non-standard module.
Convert between metric module and inch diametral pitch gear sizes.
Determine speed and torque ratio between driver and driven shafts.
Check contact ratio, undercut and tip thickness before cutting small pinions.
Calculate gear blank diameters for CNC machining.
Teach gear kinematics to mechanical engineering students.
Prototype gear sets for robotics, automation, and hobby projects.

Frequently Asked Questions

Last updated: 2026-10-09 · Reviewed by Nham Vu