Internal/External Gear Calculator

Compute all essential gear dimensions: pitch diameter, addendum/dedendum diameters, base diameter, circular pitch, center distance, and minimum teeth to avoid undercut. Based on ISO 6336 & AGMA 2000. Visualize pitch, addendum, dedendum and base circles.

Standard ext: m=2.5, z=24
Small ext: m=1, z=18
Internal ring: m=3, z=40
High PA: m=2, z=28, α=25°
Fine module: m=0.8, z=32
Local computation – All calculations run inside your browser, zero data transfer.

Gear Engineering: Theory & Practice

The involute gear profile is the foundation of modern mechanical power transmission. Developed by Leonhard Euler and later standardised by AGMA (American Gear Manufacturers Association) and ISO 6336, involute gears offer constant velocity ratio, low sensitivity to center distance errors, and ease of manufacturing. This calculator implements fundamental relations for both external (spur) gears and internal (ring) gears.

Fundamental Gear Law: The angular velocity ratio remains constant if the common normal of the tooth profiles always passes through a fixed pitch point.
Key Formulas (External):
• Pitch diameter $$ \( d = m \cdot z \) $$
• Addendum diameter $$\( d_a = m \cdot (z + 2h_a^*) \)$$
• Dedendum diameter$$ \( d_f = m \cdot (z - 2h_a^* - 2c^*) \)$$
• Base diameter$$ \( d_b = d \cdot \cos\alpha \)$$
Internal gear variations:
• $$\( d_a = m \cdot (z - 2h_a^*) \)    • \( d_f = m \cdot (z + 2h_a^* + 2c^*) \)$$
• Center distance (external pair) $$\( a = \frac{m(z_1 + z_2)}{2} \)$$
• For internal-external mesh: $$\( a = \frac{m(z_{\text{ring}} - z_{\text{pinion}})}{2} \)$$

Why Internal vs. External?

External gears are the most common configuration, used in parallel shaft transmissions. Internal gears (ring gears) allow compact planetary systems, higher load sharing, and reversed rotation direction without an idler. Common applications include automatic transmissions, epicyclic drives, and industrial robotics. Understanding the geometry ensures proper root clearance and avoids tip interference, especially important for internal meshes where the addendum circle of the pinion must not contact the ring gear's root.

Parameter Reference & Standards

Parameter Symbol Typical Values / Notes
Module m m [mm] 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10 (ISO 54)
Pressure Angle α α 20° (most common), 14.5°, 25°
Addendum coefficient ha* 1.0 (standard full-depth teeth)
Clearance coefficient c* 0.25 (standard) or 0.35 for coarse pitch
Minimum teeth (no undercut) zmin ≈ 2 / sin²α → for 20° → z_min = 17 (theoretical)
Case Study: Planetary Gearbox Design

In a typical planetary stage, a sun gear (external) meshes with planet gears (external) which also mesh with a ring gear (internal). Using our calculator, engineers can instantly verify if the ring gear's addendum diameter, root diameter, and base circle satisfy interference conditions. For module 2.5, sun z=24, ring internal z=72, the center distance between sun and planet = 60 mm, while ring internal mesh requires sufficient clearance. The tool predicts root clearance and allows rapid design iteration.

Real‑World Applications & Reliability

  • Automotive transmissions: Precise gear geometry ensures low noise and high fatigue life (AGMA 2101).
  • Robotic actuators: Harmonic drives and strain wave gearing rely on precise internal gear profiles.
  • Aerospace: Helical and spur gears in APUs and actuation systems demand ISO 6336 validation.
  • Industrial machinery: Custom gear cutting (hobbing, shaping) depends on accurate addendum/dedendum values.

Engineering rigor – This calculator implements DIN 3990 / ISO 6336‑1:2019 fundamental formulas. The internal gear formulas are validated through standard machine design textbooks (Shigley’s Mechanical Engineering Design, Juvinall & Marshek). All equations are solved analytically without approximations, giving reliable results for both standard and non‑standard coefficients. Our team has verified the results with commercial gear design suites (KISSsoft, MITCalc). Last algorithmic verification: May 2026.

Frequently Asked Questions

For a 20° pressure angle, the theoretical minimum without profile shift is 17 teeth. For 14.5° it is 32 teeth. The calculator shows a safe integer ceiling based on your actual pressure angle and addendum coefficient.

This version focuses on spur (straight teeth) involute gears. For helical gears, transverse module and helix angle must be accounted. However, many transverse-plane relations are similar. A dedicated helical tool is planned.

Addendum is the radial distance from pitch circle to tooth tip; dedendum is from pitch circle to root. Their sum equals whole depth. Standard addendum = m, dedendum = 1.25m (with clearance). For internal gears, addendum is subtracted from pitch diameter.

The center distance assumes standard (uncorrected) gears without profile shift. For most designs it provides a solid baseline. For custom center distances, profile shift (X‑factor) is required – not included in this version but noted for advanced designs.
References: AGMA 2001-D04 (Fundamental Rating Factors), ISO 6336-3:2019, “Gear Geometry and Applied Theory” by Faydor L. Litvin, and Machinery's Handbook 31st edition.
AGMA official standards | Involute Gear Wikipedia