What is the equation of the Involute of a Circle?

Parametric equations: x = r(cos t + t sin t), y = r(sin t - t cos t). r is the base circle’s radius (r = 1 in the graph) and t is the angle of string unwound.

What is the arc length of the Involute of a Circle?

Arc length (from the cusp to t): L = (r t^2)/2. r is the base circle’s radius (r = 1 in the graph) and t is the angle of string unwound.

Spirals

Involute of a Circle

The path of a string end unwinding from a circle — the profile that makes modern gear teeth run smoothly.

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Formulas and key values

Parametric equations: $x = r(\cos t + t\sin t)$ $y = r(\sin t - t\cos t)$
Arc length · from the cusp to tIn the graph ≈ 123.4: $L = \frac{r t^2}{2}$

r is the base circle’s radius (r = 1 in the graph) and t is the angle of string unwound.

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01 · Definition

What is the Involute of a Circle?

Wrap a taut string around a circle, then unwind it while keeping it tight: the free end traces the involute of the circle. At every instant the unwound string segment is tangent to the circle and perpendicular to the curve, and its length equals the arc it just left — geometry you can read directly off the parametric form x = r(cos t + t·sin t), y = r(sin t − t·cos t).

Christiaan Huygens introduced involutes in 1673 while designing pendulum clocks. Leonhard Euler later proposed the curve for gear teeth, and it stuck: two involute profiles always meet along a fixed straight line of action, so the speed ratio stays perfectly constant even when the distance between gear centers drifts.

02 · Properties

Key properties

The unwound string is always tangent to the base circle and normal to the involute.
Arc length from the starting cusp to parameter t: r·t²/2.
The curve starts at a cusp on the circle and spirals outward forever.
Its evolute (envelope of normals) is the base circle itself.
Gear meshing: contact between two involutes travels along a straight “line of action”, giving a constant transmission ratio.

03 · Questions