Last updated: July 31, 2026
Involute Function Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
inv(α) = tan(α) − α, with α in radians
Where:
- α=Pressure angle(radians)
- α_{deg}=Pressure angle(degrees)
- inv(α)=Involute function value(radians)
Worked Examples
Standard 20-degree pressure angle
A common design angle in modern involute gears.
- 1Convert 20° to radians: α ≈ 0.349066.
- 2Compute tan(α) ≈ 0.363970.
- 3Subtract α to get inv(α) ≈ 0.014904 rad.
Classic 14.5-degree pressure angle
Older gear sets often use this smaller pressure angle.
- 1Convert 14.5° to α ≈ 0.253073 rad.
- 2Compute tan(α) ≈ 0.258618.
- 3Subtract α to get inv(α) ≈ 0.005545 rad.
Check a larger first-quadrant angle
The function rises faster as the angle approaches 90°.
- 1Convert 35° to α ≈ 0.610865 rad.
- 2Evaluate tan(α) ≈ 0.700208.
- 3The involute becomes inv(α) ≈ 0.089343 rad.
Reject an invalid boundary angle
Zero degrees does not satisfy the calculator rule.
- 1The calculator requires a positive first-quadrant angle.
- 20° is outside the allowed input range.
- 3An error is returned instead of a misleading output.
Introduction
The Involute Function Calculator converts a pressure angle into the involute value used in gear geometry, standards tables, and profile calculations. It works by changing degrees to radians first, because the involute formula uses α in radians. You also get the involute result in both radians and degrees for easier comparison with design references.
What the involute function is
The involute function measures the difference between the tangent of an angle and the angle itself.
Formula is inv(α) = tan(α) − α.
The angle α must be in radians inside the formula.
Values are small for common gear pressure angles.
The function grows faster at larger angles.
It is standard notation in gear design tables.
Why degree-to-radian conversion matters
Most engineering input angles are typed in degrees, but trigonometric formulas here require radians.
Convert with α = degrees × π / 180.
20° becomes about 0.349066 rad.
14.5° becomes about 0.253073 rad.
Using degrees directly would give the wrong result.
The calculator shows the converted angle as a helper output.
How the calculator computes inv(α)
Once the angle is in radians, the calculator evaluates tan(α) and subtracts α.
Start with the converted angle.
Find tan(α).
Subtract α from tan(α).
Round the final value to six decimals.
Convert the involute value back to degrees if needed.
Valid input range
The calculator accepts only positive first-quadrant angles because tan(90°) is undefined and zero is excluded by the rule.
Angle must be greater than 0°.
Angle must be less than 90°.
Blank or non-numeric values are rejected.
The first quadrant keeps tan(α) finite.
Typical design values are well within the allowed range.
Why engineers use involute values
Involute functions appear in tooth thickness, base circle, and profile shift calculations for gears.
Helps compare standard pressure angles.
Used in gear geometry reference tables.
Supports profile inspection calculations.
Useful in machine design coursework.
Appears in CAD and handbook workflows.
How to interpret the outputs
The main result is the involute value in radians, with supporting outputs that make cross-checking easier.
invRadians is the primary engineering value.
invDegrees is convenient for reporting.
pressureAngleRadians confirms the conversion.
Larger input angles give larger involute values.
Compare your value with handbook tables when needed.
Common mistakes to avoid
Errors usually come from unit confusion or choosing an angle too close to the tangent singularity.
Typing 90° exactly, where tan is undefined.
Using degrees directly in manual formulas.
Confusing involute value with the pressure angle itself.
Rounding too early while checking tables.
Ignoring the difference between radian and degree outputs.
FAQs
What is the involute function?
It is the expression tan(α) − α, commonly used in gear geometry when α is measured in radians.
Why does the calculator ask for degrees but use radians?
Pressure angles are usually entered in degrees, but the formula itself requires radians, so the calculator converts automatically.
What are common pressure angles?
14.5° and 20° are common gear pressure angles, although the calculator accepts any valid angle between 0° and 90°.
Why is 90° not allowed?
tan(90°) is undefined, so the involute function cannot be evaluated at that angle.
Why is the involute value so small?
For typical gear angles, tan(α) and α are numerically close, so their difference is small.
What does invDegrees mean?
It is the involute value converted from radians into degrees for easier comparison with angle-based references.
Can I use this for non-gear trigonometry problems?
Yes, as long as you specifically need the involute function tan(α) − α for a first-quadrant angle.