The Spur Gear Calculator computes fundamental gear tooth geometry and circle diameters for standard external metric spur gears based on the module system (m) and pressure angle (α).

### Involute Gear Tooth Formulas & Geometry Spur gear geometry is governed by standardized relationships proportional to the metric module m (pitch diameter in millimeters divided by tooth count z): - Pitch Diameter (d): The nominal diameter where opposing gear teeth make rolling pitch contact: $$d = m \times z$$ - Base Diameter (d_b): The origin circle from which the involute tooth curvature is generated: $$d_b = d \times \cos(\alpha)$$ - Addendum (h_a): Radial tooth height extending outward beyond the pitch circle: $$h_a = 1.0 \times m$$ - Dedendum (h_f): Radial tooth depth extending inward below the pitch circle (incorporating 0.25m root clearance): $$h_f = 1.25 \times m$$ - Tip / Outside Diameter (d_a): The outer boundary circle of the blank gear disk: $$d_a = d + (2 \times h_a) = m(z + 2)$$ - Root Diameter (d_f): The inner root trough diameter at the base of the tooth cuts: $$d_f = d - (2 \times h_f) = m(z - 2.5)$$ - Circular Tooth Thickness (s): Tooth width along the circular pitch circumference: $$s = \frac{\pi \times m}{2}$$ - Circular Pitch (p): Distance along the pitch circle from one tooth face to the corresponding point on the next tooth: $$p = \pi \times m$$

### Worked Metric Gear Calculation Consider designing a pinion gear with module **m = 2.0 mm**, **z = 24 teeth**, and a standard **20° pressure angle**: 1. **Pitch Diameter**: $$d = 2.0 \times 24 = 48.0\text{ mm}$$ 2. **Base Circle Diameter**: $$d_b = 48.0 \times \cos(20^\circ) = 48.0 \times 0.9397 \approx 45.105\text{ mm}$$ 3. **Tooth Height Parameters**: - Addendum: $h_a = 2.0\text{ mm}$ - Dedendum: $h_f = 1.25 \times 2.0 = 2.5\text{ mm}$ 4. **Blank & Root Diameters**: - Outside / Tip Diameter: $d_a = 48.0 + (2 \times 2.0) = 52.0\text{ mm}$ - Root Diameter: $d_f = 48.0 - (2 \times 2.5) = 43.0\text{ mm}$ 5. **Pitch Tooth Dimensions**: - Tooth Thickness: $s = \frac{\pi \times 2.0}{2} \approx 3.142\text{ mm}$ - Circular Pitch: $p = \pi \times 2.0 \approx 6.283\text{ mm}$

### Private In-Browser Calculation All trigonometric functions and pitch circle evaluations run synchronously in your browser's local JavaScript runtime. No CAD measurements or gear specifications are transmitted across the network.