Calculate gear geometry
Pitch, tip, root and base circle of a spur gear from module and number of teeth — optionally with centre distance and ratio for a complete gear pair.
Gear geometry calculator
Pitch, tip, root, and base circle diameters for spur gears (involute gearing) — optionally with centre distance and ratio for a gear pair.
Calculate gear dimensions
Enter the module and number(s) of teeth — results appear immediately.
Formulas
dₐ = m · (z + 2)
d_f = m · (z − 2.5)
d_b = d · cos α
a = m · (z₁ + z₂) / 2
i = z₂ / z₁
z = number of teeth [—]
α = pressure angle [°]
d = pitch circle diameter [mm]
dₐ = tip circle diameter [mm]
d_f = root circle diameter [mm]
d_b = base circle diameter [mm]
a = centre distance [mm]
i = gear ratio [—]
Standard pressure angle α = 20° per DIN 867. For special gears (e.g. helical teeth, profile shift) the effective rolling-circle diameters differ — this calculator applies to uncorrected spur (straight) gears.
Typical geometry coefficients (DIN 867)
What is the gear geometry calculator for?
When sizing spur gears, every key diameter follows from the module m and the number of teeth z. The calculator delivers the four central dimensions of involute spur gearing in a single step:
- Pitch (reference) circle d = m · z — the geometric reference circle of the gearing.
- Tip circle dₐ = m · (z + 2) — the outside diameter of the gear.
- Root circle d_f = m · (z − 2.5) — the bottom of the tooth gaps.
- Base circle d_b = d · cos α — the generating circle of the involute (α = pressure angle).
Enter the number of teeth of the mating gear z₂ as well, and the tool calculates the centre distance a = m · (z₁ + z₂) / 2 and the ratio i = z₂ / z₁ — ideal for the quick pre-sizing of a gear pair. The formulas apply to uncorrected spur gearing per DIN 867 with a standard pressure angle α = 20°. For helical teeth or profile shift, the effective operating pitch circles differ.
How to determine the right module is shown in the guide Calculate gear module. Unsure which gear type to use? The gear type selector guides you to the right recommendation in a few steps.
Frequently asked questions
Which gear type does the calculator cover?
Uncorrected involute spur gearing per DIN 867 with a pressure angle of α = 20°. For helical teeth, profile shift or special gearing, additional corrections apply — the values shown here serve as a reliable pre-sizing.
How do I calculate the centre distance of two gears?
For standard gearing, a = m · (z₁ + z₂) / 2. Both gears must share the same module and pressure angle. Enter z₂ in the calculator and the centre distance appears automatically.
What is the difference between the pitch circle and the operating pitch circle?
The pitch (reference) circle d = m · z is a purely geometric property of the single gear. The operating pitch circle is the actual rolling circle in mesh. At the standard centre distance without profile shift, the two are identical.
Note: The calculation tools provided are for initial guidance only and do not replace a binding design by qualified specialists. All results must be verified by suitable engineering calculations before use in safety-relevant applications. Technische Antriebselemente GmbH accepts no liability for damage arising from the use of the calculation results.
Related products & guides
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Gears to drawing — module 0.3–8, DIN 867/3960, gear quality up to grade 5. Spur, helical and special gearing.
ProductSpur gears
Straight- and helical-cut spur gears as catalogue and custom parts — matched directly to the calculated geometry.
TutorialCalculate gear module
Determine the module from tooth count, pitch circle or centre distance — step by step with the standard module series.
ComparisonHelical vs. spur gears
Smoothness, load capacity and axial force compared — which gear type fits when.
CalculatorGear type selector
Find the right gear type in a few steps — by load, speed, noise and shaft arrangement.
GlossaryModule (gearing)
Definition and meaning of the module as the central gearing parameter — with the standard module series.