Pitch Diameter Calculator

Pitch Diameter Calculator finds the pitch diameter of a spur gear, a screw thread, a roller chain sprocket or a timing belt pulley. It also gives the outside diameter of each part.

What the Pitch Diameter Calculator Measures

Pitch diameter is the one size on a gear, thread, sprocket or belt pulley that calipers cannot read. It is the circle where the part actually works. On a gear it is where two gears roll together. On a thread it is where the ridge and the groove are equal in width.

The Pitch Diameter Calculator covers those four parts, and each one uses a different rule. The table shows the rule, the default example, and which way the pitch diameter sits against the outside of the part.

Thread Pitch Diameter and Class Limits

For a 60° Unified or ISO metric thread, the basic pitch diameter is the major diameter less 0.6495 times the pitch. The constant is 0.75 times the cosine of 30°. Some charts use 0.866 in its place, which is the full height of the sharp thread triangle and gives a wrong answer.

$$E=d-0.6495P$$

A 1/4-20 thread has a pitch of 0.05 in. Its basic pitch diameter is 0.25 less 0.0325, or 0.2175 in. The 0.866 shortcut would give 0.2067 in, which is 0.0108 in too small and well outside any class of fit.

Basic size is not the size to cut a screw to. In the Unified system it is the smallest a nut may be, and a Class 2A screw must sit below it by an allowance. The limits below are from the American Fastener Class 2A table and its Class 2B companion.

So a 1/4-20 screw is good anywhere from 0.2127 to 0.2164 in, a window of 0.0037 in. Set Part Type to Screw thread and the Pitch Diameter Calculator shows these limits beside the basic size. Each listed thread gets its Class 2A screw limits and its Class 2B nut range. Metric threads show the 6g screw and 6H nut limits. The chart under the results lists the screw limits for every size, and a few are shown here.

A thread that is not in the list, such as 1/2-12, gets a basic size but no limits. Do not cut a screw to the basic reading in that case. Take the allowance and tolerance from ASME B1.1 for the class you need.

The pitch diameter also sets the blank for a rolled thread. CJWinter, a thread rolling toolmaker, puts the blank at the largest pitch diameter less 0.002 in. For a 1/4-20 Class 2A thread that is 0.2144 in, far under the 0.25 in stock a cut thread starts from.

Three-Wire Readings for a Thread

A micrometer cannot touch the pitch line, so the thread is measured over three wires. Two wires go in the grooves on one side and one on the other. The reading converts to pitch diameter with the wire size and the pitch.

$$M=E-0.866P+3W$$

M is the reading over the wires and W is the wire diameter. The best wire touches the flanks at the pitch line. Its diameter is 0.57735 times the pitch, which is 0.0289 in for 20 threads per inch. With that wire the reading is the pitch diameter plus 0.0433 in.

A lathe operator cutting this screw should stop between 0.2560 and 0.2597 in. A reading of 0.2608 in means the thread is still 0.0011 in too large for a Class 2A fit. Type your own reading into the Pitch Diameter Calculator, under Mating Part and Measurement.

The result works it back to a pitch diameter and marks it as within limits, or states how far it is over or under. The reading presets load the largest, middle and smallest Class 2A values.

The thread result also lists the screw minor diameter, the tap drill for a 75% thread and the compound infeed at 29.5°. For 1/4-20 those are 0.1887 in, 0.2013 in and 0.0352 in. For the wider set of thread dimensions use the Thread Calculator. The Bolt Area Calculator turns the size and pitch into the stress area that carries load.

Pitch Diameter of a Gear

A gear's pitch diameter comes straight from its tooth count, and it is the first figure the Pitch Diameter Calculator shows for a spur gear. Inch gears divide the teeth by the diametral pitch. Metric gears multiply the teeth by the module.

$$D=\frac{N}{P_d}$$

$$D=mN$$

The harder job is usually the reverse, which is naming a gear you have in your hand. Count the teeth and measure across the tips. The diametral pitch is the tooth count plus two, divided by that outside diameter. A 24 tooth gear that measures 3.25 in is 26 ÷ 3.25, or 8 DP, and its pitch diameter is 3.0000 in.

Inch and metric gears can look alike and still not run together. A 24 tooth, 10 DP gear measures 66.04 mm across the tips. A 24 tooth, module 2.5 gear measures 65.00 mm. The tooth sizes differ by 1.6%, which is enough to bind. When a gear will not match a whole diametral pitch, switch Units to metric and test it as a module.

Two more results matter when a gear is being made or mounted. A standard 20° gear needs at least 17 teeth to cut without undercut, and the tool flags a count below that. The center distance is half the sum of the two pitch diameters, so a 24 and a 48 tooth gear at 8 DP sit 4.5000 in apart. Tooth proportions follow the standard full-depth form in the KHK Gear Technical Reference.

Sprocket Pitch Diameter Calculator Rule

A chain does not bend around a sprocket in a smooth circle. It lies on it as a polygon, with one flat side per tooth. The pitch diameter is therefore found with a sine, as USA Roller Chain gives it.

$$D=\frac{P}{\sin (180^{\circ }/N)}$$

The common shortcut treats the chain like a belt and uses teeth times pitch divided by π. It always comes out small, and the error is worst on small sprockets. The figures below are for #40 chain with a 0.5 in pitch.

That same polygon is why chain speed pulses. It rises and falls by 6.03% per tooth on a 9 tooth sprocket and by 0.86% on a 24 tooth one. A sprocket pitch diameter calculator that reports this ripple shows why drives with more teeth run smoother.

To check a sprocket on the bench, measure across the tooth bottoms. That bottom diameter is the pitch diameter less the chain roller diameter, or 3.5181 in for 24 teeth on #40 chain. With an odd tooth count no two bottoms sit opposite each other, so the result gives a caliper diameter in its place. For 17 teeth it is 2.3970 in.

Timing Pulley Pitch Diameter

A timing belt bends at its cords, and the cords run above the pulley teeth. The pitch diameter is the groove count times the belt pitch, divided by π, and it is larger than the pulley you can touch.

$$D=\frac{NP}{\pi }$$

Pfeifer Industries, a pulley maker, gives the outside diameter as the pitch diameter less twice the pitch line offset. Calipers across the pulley read this smaller figure. Picking a belt profile in the Pitch Diameter Calculator loads the right offset for it.

A 24 groove L pulley has a pitch diameter of 2.8648 in and measures 2.8348 in. To identify a pulley, add the offset back to the caliper reading and divide by the groove count. Belt length follows from the two pitch diameters and the center distance. For 24 and 48 groove L pulleys at 12 in centers it comes to 100.5 teeth, so a 100 tooth belt is the nearest fit.

Can Calipers Measure Pitch Diameter?

Not directly on any of the four parts. Calipers read the tips of gear teeth, the crests of a thread, the bottoms of sprocket teeth and the face of a pulley. The pitch line lies between or beyond those surfaces.

What calipers can do is confirm it. Measure the outside size, enter the tooth count or thread in the Pitch Diameter Calculator, and compare the outside diameter it returns with your reading. If the two agree, the pitch diameter shown is the right one for your part.