The involute function calculator above evaluates inv α = tan α − α, the small transcendental function that sits at the centre of every involute gear calculation. It runs in both directions, because half the work in gear geometry is going forwards from a pressure angle to an involute value and the other half is going backwards from an involute value to the angle that produced it. The reverse direction has no closed form, so the tool solves it numerically and shows you the residual.
Arb Digital publishes it because the function is deceptively awkward. It looks like something you could do on a pocket calculator, and you can, right up until you forget that α in the second term must be in radians while your calculator is in degrees. The result is then wrong by a factor of about fifty-seven and still looks plausible. Our pitch diameter calculator handles the diameters and centre distance of a gear pair; this page handles the angle function those calculations feed on.
What This Involute Function Calculator Does
In forward mode it takes an angle in degrees or radians, converts to radians, and returns tan α − α. It reports the tangent separately so you can see the subtraction happening, because the two terms are very close together at small angles and the difference between them is the entire answer. At twenty degrees the tangent is 0.36397 and the angle is 0.34907, so five significant figures of each collapse into a result of about 0.0149. That cancellation is why gear tables carry seven decimal places.
In reverse mode it takes an involute value and finds the angle that generates it. There is no algebraic inverse, so the tool uses Newton's method with the derivative of the involute function, which is tan² α. The starting guess is the cube root of three times the target, which comes from the leading term of the series expansion and is accurate enough that the iteration converges in three or four steps across the whole practical range.
The third mode is the one that connects the function to a physical gear. On an involute flank the pressure angle at any radius r satisfies cos α = rb ÷ r, where rb is the base radius. Give the tool those two radii and it returns the angle at that radius and its involute in one step. If the working radius is smaller than the base radius the tool says so in words rather than returning the arc cosine of a number greater than one.
How to Use It
- Pick the direction from the first menu. Forward for an angle you know, reverse for an involute value you have read off a drawing or a table, radii for a point on a real flank.
- Set the unit before you type. Degrees is the default because that is how pressure angles are quoted, but the function itself works in radians and the conversion is done for you.
- Read the headline number. In forward mode it is the involute value; in the other two it is the angle in degrees.
- Check the round trip. The fourth grid cell runs the calculation backwards and reports how far it landed from the input. Anything larger than the last displayed digit means the problem is badly conditioned, not that the tool is broken.
- Use the presets to reproduce the three standard pressure angles and compare their involute values side by side in the bar panel.
The Formula and How It Is Calculated
The definition is inv α = tan α − α with α in radians. The gear manufacturer KHK's technical reference on the involute gear profile gives it in exactly that form as its equation 3.2, alongside the coordinate equations for the flank curve itself. The name comes from the geometry: the curve traced by the end of a taut string unwound from a circle is the involute of that circle, and this function is the angular offset that construction produces.
Work a case by hand. Take α = 20 degrees. In radians that is 20 × π ÷ 180 = 0.3490658504. The tangent of twenty degrees is 0.3639702343. Subtract: 0.3639702343 − 0.3490658504 = 0.0149043839, which rounds to 0.0149044. That is the value printed in every gear handbook for a twenty degree pressure angle, and it is what this page returns. Repeat for 14.5 degrees and you get 0.0055448; for 25 degrees, 0.0299753.
Notice the shape of the growth. Going from 14.5 to 20 degrees roughly triples the involute value, and going from 20 to 25 doubles it again. The series expansion explains why: inv α ≈ α³ ÷ 3 for small α, so the function is cubic near the origin. That cubic behaviour is also what makes the inverse well behaved, since the cube root of three times the value is already a good approximation to the angle.
Where the Function Actually Gets Used
Three places, and they account for nearly all of it. The first is operating centre distance. When a gear pair is mounted at a distance other than the nominal one, the pressure angle at which the teeth actually mesh changes, and the relation between the two involves the involute function on both the standard and the operating angle. You cannot solve that equation for the operating angle algebraically, which is exactly why the reverse mode of this tool exists.
The second is tooth thickness at an arbitrary radius. The thickness of a tooth measured on any circle depends on its thickness on a reference circle plus the difference of two involute values. Move outward towards the tip and the involute term grows, the thickness shrinks, and at some radius it reaches zero — that is the point at which the tooth would come to a knife edge, and it sets a hard limit on how far an addendum can be extended. Our gear ratio calculator covers the speed and torque side of a pair once the geometry is settled.
The third is profile shift, sometimes called addendum modification. Shifting the cutter changes tooth thickness and the working pressure angle, and the sum of the two shift coefficients is tied to the operating angle through the involute function again. This is how a designer avoids undercut on a small pinion or hits a non-standard centre distance without a special cutter. The thread pitch calculator handles the unrelated but frequently confused problem of screw thread pitch and lead.
Why the Involute Curve Was Chosen for Gear Teeth
The involute has a property no other practical profile matches: the line of action between two meshing involute flanks is a straight line, fixed in space, tangent to both base circles. That means the direction of the contact force does not wander as the teeth roll through mesh, and the velocity ratio stays exactly constant even if the centre distance is not exactly right. A pair of involute gears mounted slightly too far apart still transmit uniform motion; they simply do it at a slightly larger pressure angle with a little more backlash.
That tolerance to centre distance error is the reason involute teeth displaced every competing profile for power transmission. Cycloidal teeth, which survive in clocks and in some watch escapements, give lower friction near the pitch point but demand precise centres. Wolfram MathWorld's page on the circle involute gives the parametric equations of the curve, x = a(cos t + t sin t) and y = a(sin t − t cos t), which is the flank shape a hob generates.
There is a second practical consequence. Because the profile is generated by a straight-sided rack cutter rolling against the blank, a single cutter of a given module and pressure angle cuts correct teeth for every tooth count. That is a manufacturing fact worth as much as the kinematic one, and it is why 20 degrees became a near-universal standard rather than one option among many.
The Numerical Traps
Near zero the function suffers catastrophic cancellation. At one degree, tan α and α agree to six significant figures, and the involute value of about 0.0000018 is the small residue left over. Computing it in single precision would lose most of the digits. This page works in double precision throughout, which keeps roughly ten good figures even at one degree, but it is worth knowing that a spreadsheet doing the same subtraction is on the same knife edge.
At the other end, as α approaches ninety degrees the tangent runs away to infinity and the involute value grows without bound. There is no gear application anywhere near there, but the tool guards the boundary rather than printing an infinity. Past ninety degrees the tangent turns negative and the expression still evaluates to a number, which is a genuine trap: it will give you an answer that means nothing. The calculator refuses that range and says why.
The reverse direction inherits the same conditioning. Because the function is cubic near the origin, a small error in a tabulated involute value becomes a proportionally smaller error in the recovered angle at large angles but a relatively larger one at small angles. If you are working from a value quoted to four decimals rather than seven, expect the recovered angle to be uncertain in its second decimal place. Our significant figures calculator is handy for keeping track of where the significant figures went.
Arb Digital builds free engineering and maths tools that print the intermediate steps instead of a bare number.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Subtracting degrees instead of radians — tan 20° minus 20 gives −19.64, not 0.0149. The tangent is dimensionless but the angle being subtracted must be the radian measure.
- Rounding the tangent before subtracting — the answer is the difference of two nearly equal numbers, so every digit you drop from tan α is a digit lost from the result.
- Reading inv α as an angle — it is a pure number that happens to have radian dimensions in its construction. It is not the pressure angle and it is not an arc length.
- Using a four-figure table for centre distance work — operating pressure angle problems amplify small errors in the involute value, so carry seven places.
- Confusing the base circle with the pitch circle — cos α = rb ÷ r uses the base radius. The pitch circle is where the standard pressure angle is quoted, not where the involute begins.
Related Free Tools From Arb Digital
Size a gear or sprocket with the pitch diameter calculator, work out speed and torque through a train with the gear ratio calculator, set up a drivetrain with the bike gear ratio calculator, convert between angle measures with the angle converter, or evaluate the underlying trigonometry with the trigonometric functions calculator. The full free online tools hub lists every engineering and mathematics tool we publish.
Frequently Asked Questions
It is inv α = tan α − α with the angle in radians. It measures the angular offset between a point on an involute curve and the point where that curve leaves the base circle, and it appears in almost every gear geometry equation.
Because the subtraction mixes a dimensionless tangent with an angle, and only radian measure makes that combination meaningful. Subtracting a number of degrees gives a result about fifty-seven times too large in the wrong direction.
0.0149044. The tangent of twenty degrees is 0.3639702 and twenty degrees is 0.3490659 radians, and the difference between them is the involute value quoted in gear handbooks.
No. There is no closed-form expression for the angle in terms of its involute value, which is why tables were printed for a century and why this tool solves it by Newton iteration instead.
Because the answer is the difference of two nearly equal numbers. Seeing both terms makes it obvious how much cancellation is happening and why seven decimal places are needed.
Zero up to but not including ninety degrees. At ninety the tangent is undefined, and beyond it the expression still evaluates but no longer describes an involute, so the tool refuses that range.
On an involute flank the pressure angle at radius r satisfies cos α = r base divided by r. Enter both radii and the tool returns the angle there and its involute value in one step.
It balances tooth strength against undercutting on small pinions and against the radial separating force on the bearings. The older 14.5 degree standard undercuts sooner; 25 degrees carries more load but pushes harder on the shafts.
This page explains a geometric function used in mechanical design for educational purposes. Any gear or drive component intended for service should be designed and verified against the applicable engineering standard by a qualified engineer.
