Involute Function Calculator

The involute function is defined as inv alpha = tan alpha minus alpha, where alpha must be measured in radians during the subtraction. It is closely connected to the involute of a circle, the curve traced by the end of a taut string as the string unwinds from a circular base. This geometry is especially important in gear design because the working faces of most modern spur and helical gear teeth use involute profiles. An involute tooth shape allows mating gears to transmit motion with a constant velocity ratio even when the center distance varies slightly. The line of action remains tangent to the base circles, producing predictable force direction and smooth engagement. Engineers use the involute function when relating pressure angles, base-circle geometry, tooth thickness, profile shifts, and measurement dimensions. Standard pressure angles such as 20 degrees therefore appear frequently in involute calculations. Although an angle may be entered in degrees, the formula itself requires radians. The calculator converts degrees by multiplying by pi and dividing by 180, then evaluates the tangent and subtracts the radian angle. For 20 degrees, alpha is approximately 0.3490658504 radians. Its tangent is approximately 0.3639702343, so the involute value is approximately 0.0149043839. The result is dimensionless because it is the difference between two angular quantities expressed in radians. The involute value grows slowly near zero and then more rapidly as the angle approaches 90 degrees. A small-angle series begins with alpha cubed divided by three, which explains why the value is much smaller than the angle for common gear pressure angles. Tangent is undefined at odd multiples of 90 degrees, so those inputs cannot produce a finite involute value. Values very near those points can also become extremely large and sensitive to rounding. This calculator supports both degree and radian workflows to reduce conversion mistakes. Use degrees when reading a drawing or gear specification stated in familiar angular units. Use radians when a previous engineering calculation already supplies alpha in mathematical form. The displayed converted radian angle makes the substitution transparent and provides a useful audit trail. For manufacturing tolerances or standards-based gear design, retain adequate precision and apply the rounding convention required by the relevant specification.