The spur gear is the most fundamental gear type, with teeth cut parallel to the shaft axis, and it appears everywhere power is transmitted: gearboxes, machine tools, automotive transmissions and pump drives. The tool below calculates the pitch circle diameter, tip (blank) diameter, root diameter, base circle, pitch and tooth depth when you enter the module (m) and number of teeth (z). If you also enter the mating gear's tooth count, the center distance, gear ratio and contact ratio are calculated as well. The tool warns you when there is a risk of undercut. By entering a profile shift (x) you can also calculate shifted gears, and for checking on the machine it gives the span measurement over k teeth (Wk), chordal tooth thickness and measurement over pins (M).
▶ Watch the walkthrough video (11 min, English)
Video walkthrough of the calculator
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This 11-minute lesson uses the calculator end to end with the worked example m = 3, z₁ = 20, z₂ = 40: module, circular pitch and pitch circle, the dimensions of a single gear, center distance and contact ratio with a mating gear, why the undercut warning appears in a rack-cutter generating simulation, a 14.5°–20°–25° pressure angle comparison and the conversion to diametral pitch. The last part shows how to find the module of a gear from its tooth count and tip diameter. The chapter links start the video at the relevant point.
- 0:57 – What a spur gear is and where it is used
- 1:35 – Module, circular pitch and pitch circle
- 2:51 – Calculator interface
- 3:47 – Single gear (m = 3, z = 20)
- 4:59 – Mating gear and center distance
- 5:46 – Meshing animation and contact ratio
- 6:36 – Undercut warning (minimum number of teeth)
- 7:11 – Undercut in a gear cutting simulation
- 7:49 – Pressure angles compared: 14.5°, 20° and 25°
- 8:35 – Low contact ratio warning
- 9:06 – Input validation
- 9:33 – Module to diametral pitch (DP)
- 9:58 – Finding the module of a gear with a caliper
- 10:39 – Summary
What is a spur gear and where is it used?
In a spur gear the teeth are cut on a cylindrical body parallel to the axis, transmitting power between two parallel shafts. Being the easiest gear type to manufacture and measure, it is the first choice both in training and in industry. Because tooth contact starts and ends across the whole face width at once, it runs noisier than a helical gear, but it generates no axial force — which simplifies bearing design. Gearboxes with moderate peripheral speeds, idler gears, pump and crane drives are typical applications of the spur gear.
Key concepts: module, pitch, pitch circle
The module (m) is the size unit of a gear — the ratio of the pitch circle diameter to the number of teeth: m = d / z. The first condition for two gears to mesh is that their modules and pressure angles are identical. The pitch circle (d) is the theoretical circle on which two gears roll on each other without slipping; all gear calculations are based on this circle. The circular pitch (p) is the arc distance between the same-side flanks of two consecutive teeth on the pitch circle, found as p = π · m. The pressure angle (α) defines the slope of the tooth profile; today the standard value is 20°, while 14.5° may be found on older gears.
Spur gear formulas
| Quantity | Symbol | Formula | Unit |
|---|---|---|---|
| Module | m | m = d / z | mm |
| Pitch circle diameter | d | d = z · m | mm |
| Tip diameter (blank diameter) | dₐ | dₐ = d + 2m = m(z + 2) | mm |
| Root diameter | d_f | d_f = d − 2.5m | mm |
| Base circle diameter | d_b | d_b = d · cosα | mm |
| Addendum | hₐ | hₐ = 1.00 · m | mm |
| Dedendum | h_f | h_f = 1.25 · m | mm |
| Whole depth (cutting depth) | h | h = 2.25 · m | mm |
| Circular pitch | p | p = π · m | mm |
| Center distance | a | a = (z₁ + z₂) · m / 2 | mm |
| Gear ratio | i | i = z₂ / z₁ = n₁ / n₂ | – |
Worked example: m = 3, z₁ = 20, z₂ = 40
For a gear pair with a 3 mm module and 20 / 40 teeth: the pinion's pitch circle is d₁ = 20 × 3 = 60 mm and the gear's is d₂ = 40 × 3 = 120 mm. The pinion's tip diameter is dₐ₁ = 60 + 2×3 = 66 mm; the blank is turned to this diameter on the lathe. The root diameter is d_f₁ = 60 − 2.5×3 = 52.5 mm and the cutting depth is h = 2.25×3 = 6.75 mm. The center distance works out to a = (20+40)×3/2 = 90 mm and the gear ratio to i = 40/20 = 2; so if the pinion turns at 1450 rpm, the gear turns at 725 rpm. Enter the same values into the tool above to see all results including the contact ratio.
Undercut and minimum number of teeth
When the number of teeth drops below a certain limit, the cutting tool digs into the tooth root and weakens it; this is called undercut. The theoretical limit is z_min = 2 / sin²α: 17 teeth at a 20° pressure angle, 32 teeth at 14.5°. If fewer teeth are needed, the solution is profile shift (cutting with the tool withdrawn from the gear); enter x in the tool above to see the dimensions of the shifted gear. The tool checks this limit automatically and shows the smallest x that avoids undercut in its warning.
Checking tooth thickness on the machine: Wk, chordal and over-pin measurement
Once a gear is cut, tooth thickness cannot be measured directly, so three indirect measurements are used. The most common is the span measurement over k teeth (Wk): the anvils of a gear tooth micrometer bear on opposite flanks of two teeth that are k teeth apart. Because the anvils touch the flanks tangentially, the reading is not affected by errors in the tip diameter. The formula is Wk = m·cosα·[π(k − 0.5) + z·invα] + 2·x·m·sinα, where invα = tanα − α (radians). k is chosen so that contact falls near the middle of the flank; for an unshifted gear k ≈ z·α/180° + 0.5.
Chordal tooth thickness is measured with a gear tooth vernier across the chord of a single tooth at the pitch circle, with the vertical slide set to the chordal height. Measurement over pins (balls), M, is the outside dimension over two pins placed in opposite tooth spaces; for odd tooth counts the pins are not diametrically opposite, so a different formula applies. Example: for m = 3, z = 20, α = 20°, x = 0 you get k = 3 and Wk = 22.981 mm; chordal tooth thickness 4.708 mm, ideal pin diameter 5.173 mm and measurement over pins M = 67.134 mm. These are nominal values; in production they are reduced by the tooth-thickness tolerance to provide backlash.
Module or diametral pitch (DP)?
In the metric system gear size is expressed by the module; in the inch system by the diametral pitch (DP) — the number of teeth per inch of pitch circle diameter. The conversion between the two systems is DP = 25.4 / m. If a gear you measure gives an "odd" module such as 2.117, that gear is most likely DP-based (25.4/2.117 = 12 DP). The tool also shows the DP equivalent of the entered module.
Frequently asked questions
How do I find the module of a gear?
Count the teeth and measure the tip diameter with a caliper — you can use the gear module finder we built for exactly this job. For a standard spur gear the module follows from m = dₐ / (z + 2). For example, a gear with 22 teeth and a 72 mm tip diameter has m = 72/24 = 3. If the result is not close to the standard series (1 – 1.25 – 1.5 – 2 – 2.5 – 3 – 4 – 5…), the gear may be profile-shifted or DP-based.
What is required for two gears to mesh?
Their modules and pressure angles must be identical. The tooth counts may differ — the gear ratio comes from that difference.
How deep do I cut when machining a gear?
For the standard full-depth profile the whole depth is h = 2.25 × m; for a module 2 gear, for example, you plunge 4.5 mm. The blank is turned to the tip diameter (dₐ = m(z+2)), then the teeth are cut to this depth.
Why does the contact ratio matter?
The contact ratio (εₐ) is the average number of tooth pairs in contact at the same time and must always be greater than 1 for spur gears. Below 1.2 the mesh runs rough and noise increases; raising the tooth counts or lowering the pressure angle increases the ratio.
What is Wk and over how many teeth is it measured?
Wk is the measurement taken with a gear tooth micrometer spanning k teeth; it shows whether the gear has been cut to the correct tooth thickness. k depends on the number of teeth, the pressure angle and the profile shift. For an unshifted 20° gear, k = 3 for z = 20 and k = 5 for z = 40. The tool above calculates k and Wk automatically for both gears.
The tool and formulas on this page are intended for education and preliminary design; they apply to the standard ISO basic rack, for both unshifted and profile-shifted gears; dimensions are nominal and exclude tooth-thickness tolerance. Verify the results before manufacturing. When cutting the gear on a machine, you can calculate the required speed and feed values here and try the turning cycles with our CNC simulators.
Other gear calculators: helical gear calculator · rack and pinion calculator · bevel gear calculator · gear formulas for every type