How do you calculate spur gear dimensions?

Pitch diameter is d = m·z, outside diameter is m(z + 2), and root diameter is m(z − 2.5) for a standard tooth. A 24-tooth module 2 gear is 48 mm on the pitch circle, 52 mm outside, and 43 mm at the root.

What is a gear module?

Module is pitch diameter divided by tooth count, in millimeters: m = d ÷ z. It sets tooth size, so gears must share the same module and pressure angle to mesh; module 2 has a 6.283 mm circular pitch.

How do you find the center distance of two spur gears?

Add the two pitch radii: a = m·(z1 + z2)/2. Module 2 gears with 24 and 48 teeth sit 72 mm apart; profile shift changes this to the working center distance a_w.

What is the minimum number of teeth on a spur gear?

Without profile shift, about 17 teeth at 20°, 32 at 14.5°, and 12 at 25° avoid undercut (z = 2 ÷ sin²α). Smaller pinions work with a positive shift, for example x ≈ 0.30 for 12 teeth at 20°, and the generator shows undercut as it is actually cut.

Should I use a 14.5°, 20°, or 25° pressure angle?

20° is the modern default. 14.5° survives in older inch gearing and many FRC parts and runs quietly but undercuts below 32 teeth; 25° gives stronger teeth and allows smaller pinions at the cost of higher bearing loads.

What contact ratio do spur gears need?

At least 1.2, so one tooth pair always takes over before the previous pair lets go. The module 2, 24 and 48-tooth example has about 1.67; the tool warns below 1.2.

Does the generator use a true involute?

Yes. Flanks are true involutes and the root is the trochoid traced by the rack cutter, so undercut on small pinions appears as it would on a cut gear. Exported curves are arcs that stay within 0.005 mm of the true form.

Can I make FRC gears with this?

Yes. The FRC preset sets 20 DP, 14.5° pressure angle, and a 1/2 in hex bore, matching the common robot gear family. Check the plate thickness you choose against the gear width your gearbox expects.

Are laser-cut gears strong enough?

Yes, for low to moderate speed and load. Laser-cut plate parts suit prototypes, robots, fixtures, and light machinery; critical, high-speed, or high-load drives need engineering review of material, thickness, hub attachment, alignment, lubrication, and wear.

Which file formats can I download?

STEP (a solid built and validated in the browser), DXF in millimeters with lines and arcs, and SVG. You can export the mating gear too, or send both to a Fabworks quote.

Power transmission calculator

Spur Gear Generator

Generate an involute spur gear from module or diametral pitch, tooth count, and pressure angle, check the mesh and undercut, then download STEP, DXF, or SVG or order it.

Design a laser-cut spur gear

Set the pitch, teeth, and pressure angle, add a mating gear, then choose a bore and plate.

Units

Presets

Pitch system

Module

Pressure angle

Teeth

Profile shiftx · module

Backlash per gearTooth thinning

Mating gear

Preview the mesh, center distance, ratio, and contact ratio.

Mating gear teeth

Mating profile shiftx · module

Hub, holes, and plate

Bore

Bore diameter

Mounting holes

Lightening holes

Fabworks plate material

Plate thicknessMin. feature 0.050 in · 1.27 mm

Gear dimensions

Pitch diameter: 48 mm
Outside diameter: 52 mm
Root diameter: 43 mm
Base circle diameter: 45.105 mm
Circular pitch: 6.283 mm
Tooth thickness at pitch: 3.092 mm
Diametral pitch: 12.7 DP
Mating pitch diameter: 96 mm
Mating outside diameter: 100 mm
Center distance: 72 mm
Gear ratio: 2 : 1
Contact ratio: 1.675

Mesh preview

Involute flanks and trochoid root fillets are generated by rolling a basic rack through each tooth space, the way a hob cuts a gear. Dashed circles are pitch circles.

True generated tooth form

Download the manufacturing files

100% in-browser

STEP solids are built and checked in your browser. DXF and SVG drawings use lines and true arcs at full scale. Nothing is uploaded until you order.

Spur gear
⌀ 51.7 mm
Mating gear
⌀ 99.9 mm

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We generate the STEP files, add them to a new quote, and open it in a new tab where you can review material, finish, and quantity.

Quick answer

How this spur gear calculator works

A spur gear’s pitch diameter is d = m·z, where m is the module and z is the tooth count (in inches, d = z ÷ DP); a standard tooth adds outside diameter m(z + 2) and root diameter m(z − 2.5). This generator traces true involute flanks and the trochoid root fillet a rack cutter leaves, sizes the mating gear and center distance, and checks undercut and contact ratio. Files are generated in the browser and download as STEP, DXF, or SVG, or go straight into a Fabworks quote.

How to use the calculator

1Choose module in millimeters or diametral pitch in teeth per inch (m = 25.4 ÷ DP), or load the FRC 20 DP or metric module 2 preset.
2Enter the tooth count (6 to 300), pressure angle (14.5°, 20°, or 25°), and any profile shift x.
3Adjust addendum, dedendum, root fillet, and backlash allowance only when you need a non-standard tooth.
4Enter the mating gear’s teeth and shift to see center distance, ratio, contact ratio, and the animated mesh.
5Pick a Fabworks material and thickness, then add a bore (round, hex, keyed, or D-flat) and up to two hole patterns.
6Fix any undercut, pointed-tooth, contact ratio, or minimum-feature warning, then download or order the gear and its mate.

Key formulas

Spur gear formulas

Pitch diameter: d = m·z = z ÷ DP; m is the module in mm, z the tooth count, DP the diametral pitch in teeth per inch.
Module and diametral pitch: m = 25.4 ÷ DP; 20 DP is module 1.27; module 2 is 12.7 DP.
Outside (tip) diameter: da = m·(z + 2 + 2x); x is the profile shift coefficient; 0 for a standard gear.
Root diameter: df = m·(z − 2.5 + 2x); Uses the ISO 53 dedendum of 1.25m.
Base diameter: db = d·cos α; α is the pressure angle. The involute flank starts at the base circle.
Circular pitch and tooth thickness: p = π·m, s = π·m/2 + 2·x·m·tan α; s is the tooth thickness on the pitch circle before backlash.
Center distance: a = m·(z1 + z2)/2; with shift, a_w = a·cos α ÷ cos α_w; α_w comes from inv α_w = inv α + 2·tan α·(x1 + x2)/(z1 + z2), where inv α = tan α − α in radians.
Contact ratio: ε = [√(ra1² − rb1²) + √(ra2² − rb2²) − a_w·sin α_w] ÷ (π·m·cos α); ra and rb are tip and base radii. Keep ε at 1.2 or more.
Undercut limit: z_min = 2 ÷ sin²α, x_min = 1 − z·sin²α/2; About 17 teeth at 20°, 32 at 14.5°, and 12 at 25° without profile shift.

Step by step

How to draw an involute spur gear profile

1Calculate d = m·z, da = m(z + 2), df = m(z − 2.5), and db = d·cos α.
2Draw the pitch, tip, root, and base circles about the gear center.
3Plot the involute from the base circle: x = rb(cos t + t·sin t), y = rb(sin t − t·cos t), from t = 0 until the radius reaches da/2.
4Rotate the involute so the tooth is π·m/2 thick on the pitch circle; its half-angle at the base circle is π/(2z) + inv α.
5Mirror the flank about the tooth centerline to form one tooth.
6Connect the flank to the root circle with a fillet of about 0.38m, or with the trochoid a rack cutter would leave (what this generator draws).
7Thin each flank by half the backlash allowance, then pattern the tooth z times around the center.
8Add the bore and holes, and check gaps and tip lands against the minimum feature size for your plate.

Worked example

Module 2, 24 teeth meshing with 48 teeth

Inputs: m = 2 mm, z = 24, 20° pressure angle, no shift, mating gear z = 48.
Result: d = 48 mm, da = 52 mm, df = 43 mm, db = 45.105 mm. Center distance a = 72 mm, ratio 2:1, contact ratio ≈ 1.67.

d = 2 × 24, da = 2 × (24 + 2), df = 2 × (24 − 2.5), and db = 48 × cos 20°. The center distance is m(z1 + z2)/2 = 2 × 72 ÷ 2 = 72 mm.

Common questions

What to know before using the result

How do you calculate spur gear dimensions?: Pitch diameter is d = m·z, outside diameter is m(z + 2), and root diameter is m(z − 2.5) for a standard tooth. A 24-tooth module 2 gear is 48 mm on the pitch circle, 52 mm outside, and 43 mm at the root.
What is a gear module?: Module is pitch diameter divided by tooth count, in millimeters: m = d ÷ z. It sets tooth size, so gears must share the same module and pressure angle to mesh; module 2 has a 6.283 mm circular pitch.
How do you find the center distance of two spur gears?: Add the two pitch radii: a = m·(z1 + z2)/2. Module 2 gears with 24 and 48 teeth sit 72 mm apart; profile shift changes this to the working center distance a_w.
What is the minimum number of teeth on a spur gear?: Without profile shift, about 17 teeth at 20°, 32 at 14.5°, and 12 at 25° avoid undercut (z = 2 ÷ sin²α). Smaller pinions work with a positive shift, for example x ≈ 0.30 for 12 teeth at 20°, and the generator shows undercut as it is actually cut.
Should I use a 14.5°, 20°, or 25° pressure angle?: 20° is the modern default. 14.5° survives in older inch gearing and many FRC parts and runs quietly but undercuts below 32 teeth; 25° gives stronger teeth and allows smaller pinions at the cost of higher bearing loads.
What contact ratio do spur gears need?: At least 1.2, so one tooth pair always takes over before the previous pair lets go. The module 2, 24 and 48-tooth example has about 1.67; the tool warns below 1.2.
Does the generator use a true involute?: Yes. Flanks are true involutes and the root is the trochoid traced by the rack cutter, so undercut on small pinions appears as it would on a cut gear. Exported curves are arcs that stay within 0.005 mm of the true form.
Can I make FRC gears with this?: Yes. The FRC preset sets 20 DP, 14.5° pressure angle, and a 1/2 in hex bore, matching the common robot gear family. Check the plate thickness you choose against the gear width your gearbox expects.
Are laser-cut gears strong enough?: Yes, for low to moderate speed and load. Laser-cut plate parts suit prototypes, robots, fixtures, and light machinery; critical, high-speed, or high-load drives need engineering review of material, thickness, hub attachment, alignment, lubrication, and wear.
Which file formats can I download?: STEP (a solid built and validated in the browser), DXF in millimeters with lines and arcs, and SVG. You can export the mating gear too, or send both to a Fabworks quote.

Reference

Module 2 spur gear dimensions, 20° pressure angle

Standard ISO 53 proportions in millimeters: addendum 1.0m, dedendum 1.25m, no profile shift.

Teeth z	Pitch dia d	Outside dia da	Root dia df	Base dia db
12	24.00	28.00	19.00	22.553
15	30.00	34.00	25.00	28.191
18	36.00	40.00	31.00	33.829
20	40.00	44.00	35.00	37.588
24	48.00	52.00	43.00	45.105
30	60.00	64.00	55.00	56.382
36	72.00	76.00	67.00	67.658
48	96.00	100.00	91.00	90.210
60	120.00	124.00	115.00	112.763

Minimum teeth before undercut by pressure angle

For a standard full-depth tooth cut by a rack (addendum 1.0m). Profile shift x moves the cutter out so fewer teeth can be cut without undercut.

Pressure angle	2 ÷ sin²α	Commonly quoted minimum	Shift x needed for 12 teeth	Shift x needed for 10 teeth
14.5°	31.9	32	0.624	0.687
20°	17.1	17	0.298	0.415
25°	11.2	12	none	0.107

Formula

d = m·z · dₐ = m(z + 2) · d_f = m(z − 2.5) · d_b = d·cos α

Pitch diameter d is module m times tooth count z, with diametral pitch converted by m = 25.4 ÷ DP. Flanks are true involutes of the base circle d·cos α, and the root is the trochoid traced by an ISO 53 rack cutter.

Engineering references

Assumptions and limits

Default tooth proportions follow ISO 53 profile A: addendum 1.0m, dedendum 1.25m, root fillet 0.38m.
Center distance and contact ratio assume rigid, correctly aligned shafts at the working center distance.
Backlash is applied by thinning each gear tooth by the entered allowance at the pitch circle.
Exported flanks are arc approximations within 0.005 mm of the generated tooth form.
Laser-cut gears suit low to moderate speed and load; critical drives need engineering review.

Educational estimate

Use this result for learning and early design exploration. Verify safety-critical or production decisions with the governing standard, material data, real tooling, and a qualified engineer.

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