Gear Tooth Calculator

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Gear Tooth Fundamentals

A gear tooth profile is defined by a set of interrelated dimensions, all derived from the basic tooth size parameter: the module (metric) or diametral pitch (Imperial). Understanding these dimensions is necessary for specifying gears, checking mesh compatibility, and designing custom gearboxes.

The key dimensions for any involute gear tooth are:

• Pitch diameter (d): The theoretical circle on which the gear teeth mesh with a mating gear. Calculated as d = m * N (metric) or d = N / P (Imperial).
• Outside diameter (d_o): The diameter of the circle passing through the tips of the teeth. d_o = d + 2m.
• Root diameter (d_r): The diameter at the base of the tooth spaces. d_r = d - 2.5m (full depth).
• Addendum (a): Radial distance from pitch circle to tooth tip. a = 1m for standard full depth.
• Dedendum (b): Radial distance from pitch circle to root. b = 1.25m for standard full depth, providing root clearance.
• Circular pitch (p): Arc distance along the pitch circle between adjacent teeth. p = pi * m.
• Tooth thickness (t): Arc thickness of the tooth at the pitch circle. t = p/2 = pi * m / 2.

Module vs Diametral Pitch

The gear industry uses two parallel systems for defining tooth size, depending on whether dimensions are in millimeters or inches.

Module (m) is the metric standard, used throughout most of the world. It equals the pitch diameter in millimeters divided by the number of teeth. A larger module means a larger (coarser) tooth. Module has units of millimeters.

Diametral pitch (P) is the Imperial standard, common in the United States. It equals the number of teeth per inch of pitch diameter. A larger diametral pitch means a smaller (finer) tooth. Diametral pitch has units of teeth per inch.

The conversion is: Module (mm) = 25.4 / Diametral Pitch, and Diametral Pitch = 25.4 / Module (mm). Two gears must have the same module (or diametral pitch) and the same pressure angle to mesh correctly.

Spur Gear Geometry

Spur gears are the simplest and most common gear type. Their teeth are straight and parallel to the shaft axis. They work well for parallel shaft applications at moderate speeds, typically below 20 m/s pitch line velocity for acceptable noise levels.

The involute tooth profile, standard since the early 1900s, ensures a constant velocity ratio between meshing gears regardless of small center distance variations. This property makes involute gears forgiving of mounting tolerances.

Key spur gear relationships for metric full-depth involute teeth:

• Pitch diameter: d = m * N
• Outside diameter: d_o = m * (N + 2)
• Root diameter: d_r = m * (N - 2.5)
• Base circle diameter: d_b = d * cos(pressure angle)
• Circular pitch: p_c = pi * m
• Tooth thickness at pitch circle: t = pi * m / 2
• Center distance (two meshing gears): C = m * (N1 + N2) / 2

Helical Gear Calculations

Helical gears have teeth cut at an angle (the helix angle) to the gear axis. This causes teeth to engage gradually rather than all at once, producing smoother, quieter operation. Helical gears are preferred for high-speed applications and automotive transmissions.

Helical gears introduce the distinction between normal and transverse planes:

• Normal module (m_n): The module measured perpendicular to the tooth face. This is the module of the cutting tool.
• Transverse module (m_t): The module measured in the plane of rotation. m_t = m_n / cos(helix angle).
• Pitch diameter: d = m_t * N = m_n * N / cos(helix angle).
• Normal pressure angle and transverse pressure angle: tan(phi_t) = tan(phi_n) / cos(helix angle).

Helical gears produce an axial thrust force equal to the tangential tooth load times tan(helix angle). This thrust must be absorbed by shaft bearings. Herringbone (double-helical) gears cancel the axial thrust by having two opposing helix angles. Typical helix angles range from 15 to 30 degrees.

Bevel Gear Overview

Bevel gears transmit motion between intersecting shafts, most commonly at 90 degrees. The teeth form on a conical surface rather than a cylindrical one.

Key bevel gear types:

• Straight bevel: Teeth are straight and converge toward the cone apex. Used at moderate speeds and loads.
• Spiral bevel: Teeth are curved for smoother engagement and higher load capacity. Standard in automotive differentials.
• Hypoid: Similar to spiral bevel but with offset axes. Found in rear axle drives where the pinion axis is below the gear axis.

For straight bevel gears with a 90-degree shaft angle, the pitch cone angles simplify to: tan(delta_1) = N1 / N2. The face width should not exceed the cone distance divided by 3, or 10 times the module, whichever is smaller.

Worm Gear Systems

A worm gear set consists of a worm (a screw-like cylindrical gear) and a worm wheel (a gear with teeth wrapped around its circumference). Worm gear sets provide very high gear ratios in a single stage, typically from 5:1 to 100:1.

• Gear ratio: ratio = wheel teeth / number of worm starts.
• Self-locking: Worm sets with lead angle below about 5 degrees are self-locking (the wheel cannot back-drive the worm). Useful for hoists and jacks.
• Efficiency: Typically 40 to 90 percent, depending on lead angle and lubrication. Higher lead angles (more starts) give better efficiency.
• Sliding contact: Worm gear teeth slide rather than roll, requiring good lubrication and bronze or similar anti-friction material for the worm wheel.

Pressure Angle Selection

The pressure angle is the angle between the tooth contact force line of action and the pitch circle tangent. It fundamentally shapes tooth geometry, strength, and meshing behavior.

The 20-degree pressure angle is the default for nearly all modern gear designs. All gears that mesh together must share the same pressure angle. Mixing pressure angles causes interference and accelerated wear.

Standard Module Sizes

Using standard module values ensures tooling availability and replacement gear sourcing. ISO 54 defines preferred sizes.

Undercutting and Profile Shift

Undercutting occurs when the gear cutter removes material from the tooth base that should be part of the involute profile. It happens when the number of teeth is below a critical minimum for the given pressure angle.

Minimum teeth to avoid undercutting (standard full-depth):

• 14.5 degrees pressure angle: 32 teeth minimum
• 20 degrees pressure angle: 17 teeth minimum
• 25 degrees pressure angle: 12 teeth minimum

When fewer teeth are needed (common for pinions), profile shift (addendum modification) moves the cutting tool radially during manufacture. A positive shift thickens the tooth base and prevents undercutting. The profile shift coefficient (x) typically ranges from -0.5 to +0.5. The sum of shift coefficients for a meshing pair determines whether the operating center distance differs from the theoretical standard distance.

AGMA Quality Grades

The American Gear Manufacturers Association defines quality grades specifying allowable tolerances for gear tooth accuracy. Higher AGMA numbers mean tighter tolerances.

Automotive transmissions typically use AGMA 9 to 11. General industrial gearboxes are usually AGMA 6 to 8. Each quality step up roughly doubles the finishing cost.

References and Standards

• AGMA 2001-D04 - basic Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth
• Machinery's Handbook, 31st Edition - Industrial Press (tables for standard tooth proportions, minimum teeth, and correction factors)
• ISO 6336 - Calculation of Load Capacity of Spur and Helical Gears (international standard parallel to AGMA 2001)
• KHK Gear Technical Reference - Gear Knowledge (detailed formulas with worked examples)

Frequently Asked Questions

What is the module of a gear?

Module is the ratio of pitch diameter (mm) to number of teeth. It defines the tooth size in the metric system. A larger module means larger teeth. Two meshing gears must have the same module. Common values range from 0.5 mm for instrument gears to 10+ mm for heavy machinery.

What is diametral pitch?

Diametral pitch (P) is the Imperial system tooth size parameter, equal to the number of teeth per inch of pitch diameter. It is the inverse relationship to module: a larger DP number means smaller teeth. Convert with module = 25.4 / DP.

How do I calculate gear ratio?

Gear ratio = driven gear teeth / driving gear teeth. A 60-tooth gear driven by a 20-tooth pinion gives a 3:1 ratio. Output speed is divided by the ratio; output torque is multiplied by it (minus 1 to 3 percent friction losses per stage).

What pressure angle should I use?

Use 20 degrees for nearly all applications. It is the current standard and provides good tooth strength with acceptable noise. Use 25 degrees only when bending strength is the primary concern. The 14.5-degree standard is obsolete for new designs.

What is the minimum number of teeth to avoid undercutting?

17 teeth for 20-degree pressure angle, 32 teeth for 14.5-degree, and 12 teeth for 25-degree. If you need fewer teeth, use profile shift (addendum modification) to prevent the tooth root from being cut away during manufacturing.

What is the difference between spur and helical gears?

Spur gears have straight teeth parallel to the shaft. Helical gears have angled teeth that engage gradually for smoother, quieter operation. Helical gears handle higher speeds and loads but produce axial thrust requiring thrust bearings.

How do I convert between module and diametral pitch?

Module (mm) = 25.4 / diametral pitch. Diametral pitch = 25.4 / module (mm). For example, module 2 mm equals DP 12.7, and DP 10 equals module 2.54 mm.

What are addendum and dedendum?

Addendum is the tooth height above the pitch circle (1 module for standard gears). Dedendum is the depth below the pitch circle (1.25 module for standard gears). The dedendum is larger to provide clearance so the mating gear tooth tip does not jam against the root.