Two pulleys, a closed timing belt, and a desired distance between the shafts: how long should the belt be? With equal pulleys, the answer is two straight spans plus one pulley circumference. With unequal pulleys, the straight spans tilt and the belt wraps farther around the larger pulley. Both changes matter.
This calculator models an open, uncrossed, two-pulley drive with parallel shafts. It finds the pitch length from a center distance, or the center distance for a belt with a known number of teeth. The GT2 / 2GT configuration here has a fixed 2 mm tooth pitch. The results describe geometry, not belt strength or a complete drive specification.
Start with the Teeth, Not the Outside Diameter
The pitch is the distance from one tooth to the corresponding point on the next tooth, measured along the belt's pitch line. Imagine marking every tooth on an ideal, unstretched belt. If the marks are \(p=2\,\mathrm{mm}\) apart, a closed belt with \(N\) teeth has pitch length
\[L=Np.\]A 120-tooth belt therefore has \(L=120\cdot2=240\,\mathrm{mm}\) of pitch length. This is the entire loop, not the length of one straight side. The belt width is a different dimension: changing a belt from 6 mm to 10 mm wide does not change the spacing between its teeth or its geometric loop length.
Now wrap that pitch line around a pulley with \(T\) teeth. One complete turn passes \(T\) tooth pitches, so the corresponding circle has circumference \(Tp\). A circle of radius \(r\) has circumference \(2\pi r\), giving the pitch radius and pitch diameter:
\[2\pi r=Tp.\] \[r=\frac{Tp}{2\pi},\qquad D_p=\frac{Tp}{\pi}.\]For a 20-tooth, 2 mm-pitch pulley, \(D_p=40/\pi\approx12.732\,\mathrm{mm}\). This is not generally the pulley outside diameter. Tooth profiles and the position of the belt's tensile members place the pitch line at a different radius. Substituting a caliper measurement of the tooth tips into a pitch-line formula mixes two geometries and produces the wrong belt length.
Replace the Drive with Two Circles and Two Straight Lines
Call the smaller pitch radius \(r\), the larger one \(R\), and the shaft center distance \(C\), all in millimeters. The pulley order does not matter for length. Define the nonnegative radius difference
\[d=R-r.\]A taut straight belt span meets each pitch circle along a tangent: a line that just touches the circle. The radius to the touching point is perpendicular to that line. If it were not perpendicular, the line would pass through the circle rather than merely touch it.
For the two pulley bodies to sit side by side, this calculator requires the pitch circles to be separated:
\[C>R+r.\]Mathematically, external tangents already exist when \(C>R-r\), but that weaker condition allows the circles to overlap. It is not a suitable default for a physical two-shaft layout. Even \(C>R+r\) does not establish real clearance: the actual pulley teeth, flanges, belt thickness, shafts, and mounting hardware still need checking.
The Right Triangle Hidden Between the Tangents
The two radii reaching the upper tangent are parallel because both are perpendicular to that tangent. Subtract the smaller radius from the larger radius. What remains is a perpendicular offset of length \(d=R-r\). Translating the tangent span alongside the center line constructs a right triangle with:
- Hypotenuse \(C\): the distance between the pulley centers.
- Perpendicular side \(d\): the radius difference.
- Parallel side \(s\): the length of one straight belt span.
The Pythagorean relation says that the sum of the squared perpendicular sides equals the squared hypotenuse: \(s^2+d^2=C^2\). Since a length is positive,
\[s=\sqrt{C^2-d^2}.\]Let \(\alpha\) be the small angle between a straight span and the line of centers. In a right triangle, the sine of an angle is its opposite side divided by its hypotenuse. Here that gives
\[\sin\alpha=\frac dC,\qquad \alpha=\arcsin\frac dC.\]The lower span has the same length and the opposite tilt because the layout is symmetric about the line of centers. Together the straight portions contribute \(2s\).
Count the Wrapped Arcs
For equal pulleys, the belt follows exactly half of each circle. A complete turn is \(2\pi\) radians, so a half-turn is \(\pi\) radians. A radian measures an angle by the arc length it cuts out on a unit-radius circle; consequently an arc of angle \(\theta\) on radius \(r\) has length \(r\theta\).
With unequal pulleys, each tangent is tilted by \(\alpha\). The upper and lower contact radii therefore remove \(\alpha\) each from the smaller pulley's half-turn and add \(\alpha\) each to the larger pulley's half-turn. The contact angles are
\[\theta_r=\pi-2\alpha,\qquad \theta_R=\pi+2\alpha.\]The sum stays \(2\pi\), but the added angle belongs to the larger radius. The two wrapped lengths add to
\[\begin{aligned} L_{\mathrm{arcs}}&=r(\pi-2\alpha)+R(\pi+2\alpha)\\ &=\pi(R+r)+2(R-r)\alpha. \end{aligned}\]Now add the two straight spans. Substituting \(d=R-r\) and \(\alpha=\arcsin(d/C)\) gives the exact pitch-line result:
\[\boxed{\begin{aligned} L&=2\sqrt{C^2-d^2}+\pi(R+r)\\ &\quad+2d\arcsin\frac dC. \end{aligned}}\]Every term has units of length. The square root is a length, \(\pi(R+r)\) is a length, and the final term is a length multiplied by a dimensionless angle in radians. A programming language's inverse sine must therefore return radians, not degrees. Degrees are used only for the displayed wrap-angle result.
The Equal-Pulley Case Is a Useful Check
If both pulleys have \(T\) teeth, then \(R=r\), so \(d=0\) and \(\alpha=0\). The exact formula reduces to
\[L=2C+2\pi r=2C+Tp.\]With 20 teeth per pulley and a 100 mm center distance, the two straight spans contribute 200 mm and the two half-circle arcs contribute 40 mm. Thus
\[L=240\,\mathrm{mm}.\] \[N=\frac{240}{2}=120.\]Conversely, a 120-tooth belt on those pulleys gives \(C=(Np-Tp)/2=(240-40)/2=100\,\mathrm{mm}\). This simple case checks the factor of two, the pitch units, and whether a belt length has accidentally been interpreted as one side of the loop.
Why a Calculated Tooth Count May Not Be a Real Belt
The geometry allows any real length, while a closed timing belt must contain a whole number of tooth pitches. A result such as 120.4 teeth is not a fractional tooth that can be ordered. The nearby possibilities are \(\lfloor N\rfloor\) and \(\lceil N\rceil\): the whole number below and above the result.
Those belts require slightly different center distances. The calculator solves both distances and reports \(\Delta C=C_{\mathrm{belt}}-C_{\mathrm{requested}}\). A positive change means the shafts need to move farther apart; a negative change means closer together. If the shorter option would make the pitch circles touch or overlap, it is discarded.
A designation such as 240-2GT-6 commonly expresses a 240 mm pitch length, a 2GT profile, and a 6 mm width. In this example the belt has 120 teeth, not 240. Suppliers use different naming conventions, and GT family names alone do not guarantee compatible tooth profiles. Confirm pitch, profile, length, width, material, and pulley compatibility against the specific manufacturer's drawing.
Find the Center Distance of a Known Belt
For unequal pulleys, \(C\) appears both inside the square root and inside the inverse sine. Instead of pretending the equation has the same simple rearrangement as the equal-pulley case, ask whether increasing \(C\) always increases \(L\). Differentiating the exact expression gives
\[\begin{aligned} \frac{dL}{dC} &=\frac{2C}{\sqrt{C^2-d^2}} -\frac{2d^2}{C\sqrt{C^2-d^2}}\\ &=2\sqrt{1-\frac{d^2}{C^2}}>0. \end{aligned}\]The derivative is positive throughout the allowed range. Thus each feasible belt length has exactly one center distance. The limiting length at touching pitch circles is obtained by substituting \(C_{\min}=R+r\); a separated layout needs a length strictly greater than that value. As \(C\) grows, the belt length grows without bound, so every longer length has a solution.
This monotonicity makes bisection reliable. Keep a lower center distance whose belt is too short and an upper distance whose belt is long enough. Evaluate their midpoint. If its belt is too short, replace the lower bound; otherwise replace the upper bound. Each step halves the interval. After \(k\) steps, an initial interval of width \(W\) is no wider than \(W/2^k\), up to floating-point rounding.
The tool limits center distances to 100,000 mm and uses 80 bisection steps. Its displayed decimals describe the numerical geometry, not manufacturing precision. Belt tolerances and tension usually dominate the final setting.
The Familiar Approximation and Its Limits
If the radius difference is small compared with the center distance, write \(u=d/C\). Near \(u=0\), \(\sqrt{1-u^2}=1-u^2/2-u^4/8+\cdots\) and \(\arcsin u=u+u^3/6+\cdots\). Substituting these into the exact length and collecting equal powers gives
\[\begin{aligned} L&=2C+\pi(R+r)\\ &\quad+\frac{d^2}{C}+\frac{d^4}{12C^3}+\cdots. \end{aligned}\]Keeping only the first correction produces the usual open-belt approximation. Since \(D=2R\) and \(D_s=2r\), it can also be written
\[\begin{aligned} L_{\mathrm{approx}}&=2C+\frac\pi2(D+D_s)\\ &\quad+\frac{(D-D_s)^2}{4C}. \end{aligned}\]The neglected terms matter more when the pulleys differ strongly in size or sit close together. The calculator uses the exact tangent expression, so it does not need this approximation. It is included to explain the origin of a formula often found in belt-sizing worksheets.
Speed Ratio and Tooth Engagement
In one driver revolution, \(T_{\mathrm{driver}}\) teeth pass a fixed point. With no slip or tooth jumping, the driven pulley must pass the same number of teeth. If \(n\) denotes revolutions per unit time,
\[n_{\mathrm{driver}}T_{\mathrm{driver}} =n_{\mathrm{driven}}T_{\mathrm{driven}}.\] \[\frac{n_{\mathrm{driven}}}{n_{\mathrm{driver}}} =\frac{T_{\mathrm{driver}}}{T_{\mathrm{driven}}}.\]A 20-tooth driver and 60-tooth driven pulley therefore produce one third of the input speed, commonly called a 3:1 reduction. An open belt makes the shafts rotate in the same direction. Ideal torque multiplication is related to the inverse speed ratio, but real allowable torque depends on losses, tooth loading and belt ratings.
The approximate number of teeth lying within a contact arc is its fraction of a turn times the pulley tooth count:
\[T_{\mathrm{wrap}}=T\frac{\theta}{2\pi}.\]The tool reports this geometric quantity for the smaller pulley, where the wrap angle is lower. It can be fractional because a contact boundary need not fall exactly between two teeth. It is not the number of teeth sharing load equally, and it is not an approval against a manufacturer's minimum engagement requirement.
What the Geometry Does Not Decide
- Installation and tension. Provide adjustment so a closed belt can be fitted and correctly tensioned. Do not treat rounding the calculated length down as a pretension design rule.
- Physical clearance. Pitch circles are an ideal model. Check tooth tips, flanges, supports, belt thickness, and neighboring hardware using actual dimensions.
- Strength and service life. Belt width, tensile construction, tooth profile, pulley diameter, speed, acceleration, duty cycle, and installation tension affect capacity. The 6-tooth input minimum is a software limit, not a recommended minimum pulley size.
- Different routing. Idlers, backside bending, crossed belts and more than two pulleys change the path. Do not reuse this two-pulley length unchanged for those arrangements.
- Real components. Tooth pitch errors, elastic stretch, temperature and wear can shift the useful center distance. Verify the assembled drive and follow its manufacturer's tensioning procedure.