How do you calculate the outside diameter of a spur gear?

Add two modules to the pitch diameter: da = m·(z + 2), or (z + 2) ÷ DP in inches. A 24-tooth module 2 gear is 52 mm; a 60-tooth 20 DP gear is 3.100 in.

How do you calculate the root diameter?

Subtract 2.5 modules from the pitch diameter: df = m·(z − 2.5). A 24-tooth module 2 gear has a 43 mm root; fine-pitch inch standards sometimes use a 1.200 ÷ DP + 0.002 in dedendum instead, which the generator lets you set.

How do I identify the module or DP of an existing gear?

Count the teeth and measure the outside diameter, then m = da ÷ (z + 2). A 24-tooth gear measuring 52 mm is module 2; a 60-tooth gear measuring 3.100 in is 62 ÷ 3.1 = 20 DP.

Whole depth is the full tooth height from root to tip, 2.25·m for a standard tooth. Module 2 teeth are 4.5 mm deep; 20 DP teeth are 0.1125 in deep.

What is the base circle?

The base circle is where the involute flank starts, with diameter d·cos α. A 48 mm pitch diameter gear at 20° has a 45.105 mm base circle.

How is center distance calculated?

Half the sum of the pitch diameters: a = m·(z1 + z2)/2. Module 2 gears with 24 and 48 teeth are 72 mm apart.

Power transmission calculator

Spur Gear Dimensions Calculator

Calculate spur gear pitch, outside, root, and base diameters, tooth depth, and center distance in module or diametral pitch, with formulas and charts.

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

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Spur gear
⌀ 51.7 mm
Mating gear
⌀ 99.9 mm

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Quick answer

How this gear dimensions calculator works

Spur gear dimensions follow from module and tooth count: pitch diameter d = m·z, outside diameter m(z + 2), root diameter m(z − 2.5), and base diameter d·cos α. In inches the same gear uses d = z ÷ DP and OD = (z + 2) ÷ DP. The calculator shows every dimension for your gear and its mate and draws the tooth profile.

How to use the calculator

1Choose module (mm) or diametral pitch (1/in).
2Enter tooth count and pressure angle.
3Read pitch, outside, root, and base diameters, plus circular pitch and tooth thickness.
4Enter the mating gear to get center distance, ratio, and contact ratio.

Key formulas

Gear dimensions formulas

Pitch diameter: d = m·z = z ÷ DP
Outside diameter: da = m·(z + 2) = (z + 2) ÷ DP
Root diameter: df = m·(z − 2.5) = (z − 2.5) ÷ DP
Base diameter: db = d·cos α; α is the pressure angle.
Whole depth: h = 2.25·m = 2.25 ÷ DP; Addendum 1.0m plus dedendum 1.25m.
Circular pitch and tooth thickness: p = π·m, s = π·m/2; Measured along the pitch circle.
Center distance: a = m·(z1 + z2)/2 = (z1 + z2) ÷ (2·DP)
Identify an unknown gear: m = da ÷ (z + 2), DP = (z + 2) ÷ da; Count teeth and measure the outside diameter.

Worked example

20 DP, 60-tooth gear in inches

Inputs: DP = 20, z = 60, 14.5° pressure angle (module 1.27 mm).
Result: Pitch diameter 3.000 in, outside diameter 3.100 in, root diameter 2.875 in, base diameter 2.904 in.

d = 60 ÷ 20, OD = 62 ÷ 20, root = (60 − 2.5) ÷ 20 with a 1.25 ÷ DP dedendum, and base = 3.000 × cos 14.5°.

Common questions

What to know before using the result

How do you calculate the outside diameter of a spur gear?: Add two modules to the pitch diameter: da = m·(z + 2), or (z + 2) ÷ DP in inches. A 24-tooth module 2 gear is 52 mm; a 60-tooth 20 DP gear is 3.100 in.
How do you calculate the root diameter?: Subtract 2.5 modules from the pitch diameter: df = m·(z − 2.5). A 24-tooth module 2 gear has a 43 mm root; fine-pitch inch standards sometimes use a 1.200 ÷ DP + 0.002 in dedendum instead, which the generator lets you set.
How do I identify the module or DP of an existing gear?: Count the teeth and measure the outside diameter, then m = da ÷ (z + 2). A 24-tooth gear measuring 52 mm is module 2; a 60-tooth gear measuring 3.100 in is 62 ÷ 3.1 = 20 DP.
What is whole depth?: Whole depth is the full tooth height from root to tip, 2.25·m for a standard tooth. Module 2 teeth are 4.5 mm deep; 20 DP teeth are 0.1125 in deep.
What is the base circle?: The base circle is where the involute flank starts, with diameter d·cos α. A 48 mm pitch diameter gear at 20° has a 45.105 mm base circle.
How is center distance calculated?: Half the sum of the pitch diameters: a = m·(z1 + z2)/2. Module 2 gears with 24 and 48 teeth are 72 mm apart.

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

20 DP spur gear dimensions, 14.5° pressure angle (in)

Full-depth teeth with a 1.25 ÷ DP dedendum and no profile shift.

Teeth	Pitch dia	Outside dia	Root dia	Base dia
12	0.600	0.700	0.475	0.581
15	0.750	0.850	0.625	0.726
20	1.000	1.100	0.875	0.968
24	1.200	1.300	1.075	1.162
30	1.500	1.600	1.375	1.452
36	1.800	1.900	1.675	1.743
48	2.400	2.500	2.275	2.324
60	3.000	3.100	2.875	2.904

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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