Last updated October 5, 2026 · Report an error
The involute function, inv(θ) = tan(θ) - θ, appears throughout gear geometry and describes the shape traced by a taut string as it unwinds from a circle — imagine pulling a pencil attached to a string wrapped around a spool, keeping the string taut as it unrolls. That traced curve, called an involute of a circle, is exactly the profile used for the teeth of virtually all modern spur and helical gears, because involute-profile gear teeth mesh smoothly and maintain a constant velocity ratio even with small manufacturing or spacing errors. Mechanical and automotive engineers use the involute function specifically when calculating gear tooth thickness and backlash allowances.
How it works
Enter the angle θ (in degrees or radians) — this represents the "roll angle" as the imaginary string unwinds from the base circle. The calculator converts to radians if needed (since the formula is defined in radians), then subtracts θ from tan(θ) to get inv(θ) = tan(θ) - θ. Because θ must be in radians in this formula, entering degrees requires converting first (multiplying by π/180).
- Enter angle (θ).
- Enter angle unit.
- Click Calculate to see your results.
Examples
θ = 20°
θ ≈ 0.349066 rad. inv(θ) = tan(0.349066) - 0.349066 ≈ 0.363970 - 0.349066 ≈ 0.014904.
θ = 25°
θ ≈ 0.436332 rad. inv(θ) = tan(0.436332) - 0.436332 ≈ 0.466308 - 0.436332 ≈ 0.029976.
Common mistakes to avoid
- Entering an angle of 90° or 270° (or their radian equivalents), where tan(θ) is undefined.
- Forgetting to convert degrees to radians before applying the formula manually (the calculator handles this automatically based on your selected unit).
- Confusing the involute function with the general concept of an "involute curve" — the function inv(θ) is a specific numerical formula used in gear calculations, while "involute" more broadly describes the whole family of unwound-string curves.