The Complete Guide to Involute Gear Geometry
Every gear in the world, from a wristwatch pinion to a wind-turbine stage, shares the same handful of defining circles. Once you know the tooth count, pitch, and pressure angle, every other dimension — addendum, dedendum, outside diameter, base circle — follows from a few formulas. This guide walks through those circles one at a time and shows how the Gear Tooth Generator assembles them into a complete, drawable gear.
The story begins with the pitch circle, the imaginary circle on which the gear's effective radius lives. The pitch diameter D is the first thing computed: for metric gears D = N × m, where N is the number of teeth and m is the module in millimeters; for imperial gears D = N ÷ P, where P is the diametral pitch in teeth per inch. The two systems are reciprocals of each other — module is millimeters of pitch diameter per tooth, diametral pitch is teeth per inch of diameter — related by m = 25.4 ÷ P. Two meshing gears share the same pitch, and the pitch circles roll together without slipping, which is why center distance is the sum of the two pitch radii.
Around the pitch circle the teeth are spaced at the circular pitch, p = π × m, the distance from one tooth to the next measured along the pitch circle. Half of that, p ÷ 2, is the tooth thickness at the pitch circle, and the other half is the width of the space. This division is the standard full-depth system's default: tooth thickness equals tooth space, leaving no intentional backlash. Real gears add a small backlash by cutting the teeth slightly thinner, but the nominal geometry starts from this even split.
Each tooth rises above the pitch circle by the addendum a and falls below it by the dedendum b. In the standard system the addendum equals one module (a = m) and the dedendum equals 1.25 modules (b = 1.25m). The dedendum is deliberately larger than the addendum so the root of one gear clears the tip of its partner — the 0.25m gap keeps the tips from grinding the roots. The whole depth of the tooth space is the sum, 2.25m, and the outside diameter is OD = D + 2a = (N + 2)m while the root diameter is DR = D − 2b = (N − 2.5)m.
The most important circle is the one that usually gets no mention in catalogs: the base circle. The base circle diameter is Db = D × cos α, where α is the pressure angle — the standard is 20°, with 14.5° and 25° as historical and high-strength variants. The base circle matters because the involute — the shape of every working gear tooth — is generated by unwinding a taut line from it. A point on the unwinding string traces the curve that becomes the tooth flank. Because the involute is generated by pure rolling of the line against the base circle, a pair of gears cut from the same base-circle geometry always transmits motion with a constant velocity ratio, regardless of how far apart the centers are set — this is the property that makes involute gearing the universal standard.
No involute exists below the base circle, because the generating line cannot unwind past its wrapping point. Between the root and base circles the flank is therefore not involute — it is a straight radial line (or a trochoid in a generated gear). This is a detail the calculator handles in its drawing: the working flank, from the base circle out to the tip, is a true involute, and the region below is the non-active root. It is also why very small tooth counts fail: when the base circle is large relative to the gear, the involute starts above the root circle and part of the tooth is cut away, producing the weakness known as undercut.
The pressure angle describes how the flank slopes at the pitch circle and controls how the force between meshing teeth is directed. A 20° pressure angle puts a larger component of force radially than a 14.5° angle, increasing bearing loads but also increasing tooth strength and reducing interference. That is why 20° replaced 14.5° as the modern standard: it carries more load per tooth for the same material. The base circle follows directly — a steeper pressure angle means a smaller base circle and a beefier tooth — so the choice of pressure angle is really a choice about where the involute starts.
The calculator brings all of this together in one pass. Enter the teeth, the module or diametral pitch, and the pressure angle, and it computes pitch diameter, addendum, dedendum, whole depth, outside diameter, root diameter, circular pitch, base circle, and tooth thickness, printing each formula with the values substituted. Enter a meshing gear's tooth count and it adds the center distance and speed ratio. The canvas then plots the true involute flank from the base circle, the tip arcs, and the root fillets, and rotates the profile around the center — the same construction a drafter would do by hand, automated.
The practical takeaway for anyone cutting or specifying gears is the ordering of the inputs. Teeth and pitch set the size; pressure angle sets the tooth shape and strength; and the addendum and dedendum constants (1.0 and 1.25 modules) set the depth and clearance. Change any one and every diameter moves, which is why the same gear can be quoted so differently by two suppliers who disagreed on the system. Knowing the circles and the formulas — the ones on this page — lets you check any gear drawing before it is cut.
Run the generator on the gear you are specifying: confirm the pitch, check the base circle, and verify the center distance against the housing. The formulas on this page are the entire foundation of gear geometry, and the tool makes them auditable in seconds.