Gear Geometry, Types, Rating and Selection: The Language of Gears

Context and scope

The definitive, engineer-grade manual on gears — from first principles to advanced design — told through the lens of real-world engineering challenges.

The Language of Gears: Definitions That Unlock Everything

Before you can design a single gear, you need to speak the language. These definitions are not academic exercises. Each one represents a dimension, a force, or a constraint that will show up in your calculations, your drawings, and your failure analyses.

Read this section once for orientation. Return to it as a reference. Treat it like a dictionary you carry in your back pocket.

The Critical Terms

Addendum — The radial distance from the pitch circle to the top of the tooth. Think of it as "how far the tooth sticks up above the rolling surface."

Dedendum — The radial distance from the pitch circle to the bottom of the tooth space. The dedendum is always slightly larger than the addendum of the mating gear to provide clearance.

Pitch Circle — The imaginary circle on which the gear effectively "rolls" against its mate. When two gears mesh, their pitch circles are tangent to each other. This is the reference circle for nearly every calculation you will ever perform on a gear.

Pitch Diameter (D) — The diameter of the pitch circle. For a standard spur gear:

D = N ÷ P

Where N = number of teeth and P = diametral pitch.

Diametral Pitch (P) — The number of teeth per unit of pitch diameter. The higher the number, the finer (smaller) the teeth. A gear with P = 4 has large teeth; a gear with P = 48 has tiny teeth.

Circular Pitch (p) — The distance along the pitch circle from one tooth to the corresponding point on the next tooth.

p = π ÷ P

Module (m) — Used in metric-based systems. The module is the pitch diameter in millimeters divided by the number of teeth. It is the reciprocal concept of diametral pitch.

m = D (mm) ÷ N

Base Circle — The circle from which the involute tooth profile is generated. Every involute gear has a base circle, and its diameter determines the shape of the tooth.

Base Circle Diameter:

D_B = D × cos(φ)

Where φ is the pressure angle.

Pressure Angle (φ) — The angle between the line of action (the path along which tooth contact occurs) and a line perpendicular to the line connecting the gear centers. Standard values are 14.5°, 20°, and 25°.

This is the single most important angle in gear design. Get it wrong, and everything downstream fails — just as the practitioner learned.

Working Depth — The depth of engagement between two mating gears. Equal to the sum of their addendums.

Whole Depth — Total depth of the tooth space. Equals the working depth plus clearance.

Clearance — The radial distance between the top of one tooth and the bottom of the mating tooth space. Prevents metal-to-metal contact at the root.

Backlash — The shortest distance between the non-driving tooth surfaces when the working flanks are in contact. It is the "play" between gears. Too little creates overheating and binding. Too much creates impact loading and noise.

Face Width — The dimension of the tooth measured parallel to the gear axis. A wider face can carry more load, but too wide relative to the pitch diameter causes uneven load distribution.

Contact Ratio — The average number of teeth in contact at any moment. Must be well above 1.0 (typically 1.4 or higher for power transmission) to ensure smooth transfer of load from one pair of teeth to the next.

Involute — The curve used as the profile of gear teeth. It is the path traced by the end of a taut string unwinding from the base circle.

Line of Action — The straight line tangent to both base circles along which tooth contact occurs. This is where the forces are transmitted.

Gear Ratio (m_G) — The ratio of the number of teeth in the driven gear to the number in the driving gear.

m_G = N_G ÷ N_P

Anatomy Diagram: The Gear Tooth

        ┌─────────── Top Land ────────────┐
        │                                  │
  ──────┤         TOOTH TIP                ├──────  ← Outside Circle (O.D.)
        │                                  │
        │←─── Addendum ───→│               │
  ------│------ PITCH LINE -│---------------│------  ← Pitch Circle
        │                   │               │
        │←── Dedendum ────→│               │
        │                   │               │
  ──────┤     TOOTH ROOT    │  TOOTH SPACE  ├──────  ← Root Circle
        │                   │               │
        └───────────────────┘               │
              ↑                             │
            Fillet Radius              Bottom Land

  ←──── Circular Pitch (p) ────→

  ←─ Tooth    →←── Space ──→
    Thickness      Width

Gear Tooth Cross-Section with Pressure Angle

                    │
                    │  Line of Action
                   ╱│
                  ╱ │
                 ╱  │  ← Pressure Angle (φ)
                ╱   │
               ╱    │
    ──────────╱─────┼──────────  Perpendicular to
             ╱      │              center line
            ╱       │
           ╱        │
    Driving         │        Driven
    Gear            │        Gear
                    │
         ← Center Distance →

The Involute Curve: The Geometry That Changed the World

The Nine Rules of Involute Gears

These are the foundational principles. Every gear calculation you ever perform rests on these:

1. The shape of an involute curve depends only on the diameter of the base circle from which it is derived.

2. Involute gears transmit uniform angular motion even when center distance varies.

3. The relative rate of motion between mating gears is established by the diameters of their base circles — not their pitch circles.

4. Contact between meshing involute teeth is along a straight line tangent to both base circles. This is the line of action.

5. The pitch circles are determined by the intersection of the line of action and the common center line. Thus, pitch diameters change with center distance.

6. Pitch diameters are directly proportional to base circle diameters.

7. The pressure angle is the angle between the line of action and a line perpendicular to the common center line. Pressure angle changes with center distance.

8. When an involute gear meshes with a rack, the rack's straight tooth flanks are tangent to the involute and perpendicular to the line of action.

9. An involute pinion driving a straight-sided rack at uniform speed produces uniform linear rack motion.

Visualizing the Involute Generation

        Base Circle
       ╱           ╲
      │      ○       │  ← Center
      │     ╱│       │
       ╲   ╱ │      ╱
        ╲ ╱  │     ╱
     ────╳───┼────    ← String (always tangent to base circle)
        ╱ ╲  │  ╲
       ╱   ╲ │   ╲
      ╱     ╲│    ╲
             ●     ╲   ← Pencil traces the involute curve
            ╱       ╲
           ╱         ╲
    ──────╱───────────╲──── Involute Curve

Spur Gears: The Workhorse of Mechanical Engineering

What They Are

External spur gears are cylindrical gears with straight teeth cut parallel to the axis. They transmit drive between parallel shafts and are the simplest, most common type of gear in existence.

When to Use Them

• Parallel shafts rotating in opposite directions
• Moderate speeds where maximum smoothness is not critical
• Applications where no axial thrust is acceptable
• Cost-sensitive designs requiring simple manufacturing

When NOT to Use Them

• High-speed applications where noise is a concern (spur gears tend to be noisy at high speeds because the entire tooth engages simultaneously)
• Applications requiring same-direction rotation on parallel shafts (use internal spur gears for this)

Internal Spur Gears

Internal spur gears have teeth on the inner cylindrical surface. They provide compact drive arrangements for transmitting motion between parallel shafts rotating in the same direction.

The Master Formula Table for Spur Gears

This is the table you will use more than any other in your career. Print it. Laminate it. Tape it to your desk.

Tooth Proportions by System

Critical note: The 14.5-degree full-depth system and the 20-degree stub system are not recommended for new designs. The industry standard is the 20-degree full-depth involute system per ANSI B6.1-1968.

Contact Ratio: The Smoothness Guarantee

The contact ratio tells you, on average, how many pairs of teeth are in mesh at any moment. If the contact ratio drops below 1.0, there are instants when no teeth are in contact — and the gear pair will impact, vibrate, and fail.

The formula:

m_f = Length of Action ÷ Base Pitch

Rules of thumb:

• Minimum for power gears: 1.4
• Below 1.2: Not recommended except in extreme, controlled cases
• Above 2.0: Excellent smoothness, but requires careful profile design

Case Study: The 7-Tooth Pinion Problem

Consider an illustrative engineering practitioner who needs to fit a gear reducer into a space so tight that the pinion must have only 7 teeth. With standard 20-degree, full-depth proportions, a 7-tooth pinion would suffer severe undercut — the cutting tool would gouge away the base of the tooth, destroying the involute profile and drastically weakening the tooth.

The solution: long-and-short addendum design. The pinion gets an enlarged addendum (the tooth extends further from the pitch circle), while the mating gear gets a shortened addendum. This eliminates undercut, preserves the contact ratio, and maintains the correct center distance.

The ANSI standard provides precise tables for these enlarged pinion dimensions, covering pinions with as few as 7 teeth meshing with gears of up to 400+ teeth.

The takeaway: Standard proportions are a starting point, not a prison. When the application demands it, modified proportions — calculated precisely — deliver solutions that standard gears cannot.

Formulas for Outside and Root Diameters

For gears that are finish-hobbed, shaped, or pre-shaved, the exact formulas depend on both the tooth form and the manufacturing process:

20-degree Full-Depth, Hobbed:

• Outside Diameter: D_O = N/P + 2/P
• Root Diameter: D_R = N/P − 2.314/P

20-degree Full-Depth, Shaped:

• Root Diameter: D_R = N/P − 2.5/P

20-degree Full-Depth, Pre-shaved:

• Root Diameter: D_R = N/P − 2.7/P

Fine-Pitch (20P and finer), Hobbed or Shaped:

• Root Diameter: D_R = N/P − 2(1.2/P + 0.002)

Why the difference? Each manufacturing method leaves a different fillet shape at the root. Shaping and pre-shaving produce a higher fillet trochoid, requiring more clearance to avoid interference with the mating tooth tip.

Helical Gears: When Silence and Strength Collide

The Scene

Picture a wind turbine gearbox. It sits 80 meters above the ground, inside a nacelle that must remain as quiet as possible to comply with local noise regulations. The gearbox converts the slow rotation of the rotor (10–20 RPM) to the 1,500 RPM the generator demands. The loads are enormous, variable, and never truly constant.

Spur gears would work, mechanically. But the noise would be unacceptable. The solution: helical gears.

What Makes Helical Gears Different

Helical gears have teeth cut at an angle to the axis. This angle — the helix angle (α) — means that tooth engagement is gradual rather than sudden. Instead of the entire tooth face contacting at once (as in spur gears), contact begins at one end and sweeps across the face.

The result: superior load-carrying capacity and dramatically reduced noise compared to spur gears of the same size.

The Tradeoff: Axial Thrust

The angled teeth produce a force component parallel to the shaft axis — axial thrust. This thrust must be absorbed by thrust bearings, adding cost and complexity to the design. The magnitude of the thrust increases with the helix angle.

Axial Thrust = Tangential Force × tan(α)

The Helical Gear Master Formulas

Where:

• P_n = normal diametral pitch of the cutter
• α = helix angle
• N = number of teeth

Direction of Thrust: A Design Decision

The first step in helical gear design is determining the desired direction of thrust. Once you know:

1. The desired thrust direction
2. The relative positions of driver and driven gears
3. The direction of rotation

...you can determine whether the helix should be right-hand or left-hand.

Rule: The thrust direction can be reversed by changing any one of these three conditions: the hand of the helix, the direction of rotation, or the driver/driven assignment.

Crossed Helical Gears

When helical gears mesh on non-parallel axes, they become crossed helical gears. These are used for light loads and low-precision applications where the shafts are skewed relative to each other.

Helical Gear Tooth Geometry

        ┌──────────────────────────────┐
        │            FACE WIDTH         │
        │  ╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱  │
        │ ╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱  │  ← Teeth at helix angle α
        │╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱  │
        │╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱   │
        │╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱╱    │
        └──────────────────────────────┘
                     │
              ←───── α ─────→  (Helix Angle)
                     │
               Gear Axis Direction

Bevel Gears: Turning Corners in Power Transmission

The Challenge: Intersecting Axes

When power must be transmitted between shafts that intersect — typically at 90 degrees, but also at acute or obtuse angles — bevel gears are the answer. Where spur and helical gears work with parallel shafts, bevel gears are built on cones instead of cylinders.

Types of Bevel Gears

Straight Bevel Gears

Teeth are radial toward the cone apex and are of conical form. The simplest bevel gear to manufacture and calculate.

• Best for: Peripheral speeds up to 1,000 feet per minute
• Limitations: Entire tooth engages at once (like spur gears), so noise increases at high speeds
• Thrust: End thrust under load tends to separate the gears

Spiral Bevel Gears

Teeth are curved and oblique, contacting each other gradually from one end to the other. The meshing action is smoother and quieter than straight bevel gears.

• Key detail: The hand of the spiral on the pinion is always opposite to that of the gear
• Spiral angle: Does not affect smoothness or efficiency but does affect thrust direction
• Left-hand spiral pinion driving clockwise (viewed from the large end): creates thrust that tends to move the pinion out of mesh

Zerol Bevel Gears

Curved teeth lying in the same general direction as straight bevel teeth. Think of them as spiral bevel gears with a zero spiral angle.

• Use instead of straight bevel: When hardened, high-accuracy gears are required (Zerol gears can be ground; straight bevels generally cannot on spiral-type equipment)

Hypoid Bevel Gears

A cross between spiral bevel gears and worm gears. The axes are non-intersecting and non-parallel. The distance between axes is called the offset.

• Advantage: Higher ratios than other bevel gears; the pinion can be made larger (stronger) due to the larger spiral angle
• Advantage: Sliding along the tooth length makes them even smoother than spiral bevels
• Can be ground on the same machines used for spiral and Zerol bevels

Applications Summary

Bevel Gear Nomenclature Diagram

                        VERTEX (Apex)
                       ╱ │ ╲
                      ╱  │  ╲
                     ╱   │   ╲
                    ╱    │    ╲
                   ╱ Face│Angle ╲
                  ╱ (γ)  │      ╲
                 ╱───────┼───────╲   ← Outside Diameter (O)
                ╱  Pitch │Cone    ╲
               ╱   Angle │(α)     ╲
              ╱    ──────┼─────    ╲  ← Pitch Diameter (D)
             ╱  Cutting  │Angle    ╲
            ╱   (δ = ζ)  │         ╲
           ╱─────────────┼──────────╲ ← Root
          ╱              │           ╲
         ╱               │            ╲
                         │
                    ← Vertex Distance (J) →

    Width of Face (F) is measured along the cone surface
    Pitch Cone Radius (C) = D ÷ (2 × sin α)

Milled Bevel Gear Formulas (Shafts at 90°)

For shafts at angles other than 90°:

tan α_P = sin Σ ÷ (N_G/N_P + cos Σ)

Where Σ = angle between shafts. Note: For shaft angles greater than 90°, the cosine is negative.

Mounting Matters: Bearing Spacing and Shaft Stiffness

This is where many bevel gear installations fail — not from bad gear design, but from inadequate mounting.

Rules for bevel gear mounting:

• Bearing spread must never be less than 70% of the gear's pitch diameter
• For overhung gears: bearing spread must be at least 2.5 times the overhang
• Shaft diameter should be equal to or greater than the overhang distance
• Axial thrust should be taken at one location only, near the gear with greater thrust
• Always provide for axial adjustment of both gear and pinion during assembly

Worm Gears: The Masters of Reduction

The Story of Self-Locking

an illustrative engineering practitioner faced a classic dilemma: he needed a speed reduction of 40:1 in a space that could barely fit a shoebox. The system had to be self-locking — if power was lost, the elevator car must not free-fall under the load of gravity.

Worm gearing was the only answer.

How Worm Gears Work

A worm gear set consists of a worm (a screw-like component) meshing with a wormgear (a gear with teeth shaped to wrap around the worm). The worm drives the wormgear, and the axes are at right angles in the most common configuration.

The key characteristics:

• Line contact between worm and wormgear teeth
• Ratios from 1.5:1 to 100:1 in a single stage
• Efficiency decreases as the ratio increases — this is the trade-off
• Self-locking is possible with single-thread worms at low lead angles

Fine-Pitch vs. Coarse-Pitch Worm Gearing

Single-Thread vs. Multi-Thread Worms

Single-thread worms produce high ratios but are comparatively inefficient due to the low lead angle. They are used for:

• Large speed reductions in a single stage
• Adjustment mechanisms
• Applications where self-locking or mechanical advantage is critical

Recommended maximum ratio for single worm set: 50:1 (up to 100:1 is possible but rarely practical)

Multi-thread worms are used when the primary goal is efficient power transmission. The lead angle should be 25° to 45° for best efficiency.

Example: For a ratio of 6:1, possible combinations include:

Hunting tooth action: The number of wormgear teeth should not be an exact multiple of the worm threads. This ensures even wear distribution.

Materials for Worm Gearing

The standard combination for power worm gearing:

• Worm: Hardened and ground steel
• Wormgear: Phosphor bronze (10–12% tin)

The specific alloy:

SAE No. 65 Phosphor Gear Bronze

Fine-Pitch Worm Gear Formulas (ANSI B6.9-1977)

Standard Axial Pitches: 0.030, 0.040, 0.050, 0.065, 0.080, 0.100, 0.130, 0.160 inch

Standard Lead Angles: 0.5°, 1°, 1.5°, 2°, 3°, 4°, 5°, 7°, 9°, 11°, 14°, 17°, 21°, 25°, 30°

Where:

• n = number of threads in worm
• P_x = axial pitch of worm
• P_n = normal circular pitch = P_x × cos λ
• λ = lead angle of worm
• N = number of teeth in wormgear
• C = center distance

Planetary Gearing: Compact Power, Infinite Ratios

The Concept

Planetary (epicyclic) gearing provides a compact design with driving and driven shafts in line and potentially very large speed reductions. The arrangement consists of:

1. A sun gear (central gear)
2. Planet gears (orbiting gears mounted on a carrier arm)
3. A ring gear (internal gear surrounding the planets)

By fixing different elements, you can create speed increases, speed reductions, or even reverse rotation.

Speed Ratio Formulas

The behavior of planetary gearing depends on which element is fixed, which drives, and which is driven.

Notation:

• A = size of driving gear (teeth or pitch diameter)
• B = size of driven gear
• C = size of fixed gear
• x, y = sizes of planet gears
• F = rotation of follower per revolution of driver
• D = rotation of driver per revolution of follower

Key principle: If the result is negative, the driver and follower rotate in opposite directions.

Common Configurations

Compound Drive: When there are two driving members rotating at different speeds, the formulas become:

F = 1 + [z × (1 − S)] ÷ B

Where z = size of secondary driving gear and S = rotation of secondary driver per revolution of initial driver. S is negative when the secondary and initial drivers rotate in opposite directions.

Planetary Bevel Gears

Two forms exist:

1. A planet gear rotates about a fixed bevel gear at the center of which is the driven shaft
2. The Humpage reduction gear — sometimes called cone-pulley back-gearing, used within cone pulleys of certain machine tools

Why Planetary Gears a desktop spreadsheet application

• Coaxial input and output — no offset between shafts
• Very high ratios in small packages
• Load sharing among multiple planet gears reduces stress on any single tooth
• Compact and lightweight relative to parallel-shaft arrangements

Herringbone and Double Helical Gears: The High-Speed Elite

Where They Dominate

Herringbone gears are the gear of choice in high-speed, high-power transmissions:

• Marine reduction gears
• Steam turbine drives
• Electric motor speed reducers
• Any transmission where pitch-line velocity ranges from 1,000 to 12,000+ feet per minute

Why Herringbone Gears Exist

Helical gears produce axial thrust. Herringbone gears eliminate it by placing two opposing helical gear sections on a single gear — one right-hand, one left-hand. The axial thrusts cancel each other perfectly.

Design Considerations

The design problem has two forms:

1. Given: Power and speed requirements. Find: Required gear proportions.
2. Given: Gear proportions and speed. Find: Power-transmitting capacity.

The first is harder and more common.

Failure Modes

Herringbone gear failures are rarely tooth breakage. Instead, they fail by:

• Excessive surface wear
• Pitting (sub-surface fatigue)
• Spalling (flaking of the hardened surface)

There exists a critical surface pressure for any given material. Above this pressure, wear is rapid and gear life short. Below it, wear is negligible.

The critical point is the yield point or endurance limit of the material.

In practical design, a reasonable factor of safety is applied below this critical value.

Ratchet Gearing: Motion in One Direction Only

What Ratchet Gears Do

Ratchet gearing serves two purposes:

1. Transmit intermittent motion — converting oscillating input to one-directional rotation
2. Prevent backward rotation — a holding mechanism (used extensively in hoisting equipment)

Types of Ratchet Mechanisms

Simple Ratchet and Pawl — A toothed wheel (ratchet) with a pivoted pawl. The pawl engages the teeth during forward motion and slides over them during return.

Stationary Pawl — The pawl is fixed; its only job is to prevent backward rotation of the ratchet wheel.

Locking Pawl — Prevents rotation in either direction while engaged.

Multiple-Pawl Ratchet — Two or more pawls of different lengths, with the length difference equal to a fraction of the tooth pitch. This effectively reduces the pitch without weakening the teeth.

Example: Two pawls differing in length by half the tooth pitch halve the effective feed per stroke while keeping the teeth strong.

Reversible Ratchet — Teeth shaped for driving from either side. A double-ended pawl swaps which face drives.

Frictional Ratchet — No positive tooth engagement. Rollers or balls wedge between the ratchet wheel and an outer ring when turned in one direction, creating a one-way clutch effect.

Linear Ratchet — Used in lifting jacks. One pawl does the lifting; another holds the load while the lifting pawl resets.

Tooth Shape: A Structural Decision

The face of each tooth must be oriented so that a line perpendicular to the engaging face passes between the ratchet wheel center and the pawl pivot center.

• If the pawl pushes the ratchet: perpendicular falls between centers
• If the pawl pulls the ratchet: perpendicular falls outside the pawl pivot center

Pitch Formula for Load-Holding Ratchets

P = √(M ÷ (L × S × N × F))

Where:

• P = circular pitch at outside circumference
• M = turning moment on the ratchet shaft (force × distance units)
• L = face width (thickness of ratchet gear)
• S = safe stress (for steel: 2,500 under shock; 4,000 without shock, in consistent pressure units)
• N = number of teeth
• F = factor (50 for ≤12 teeth; 35 for 12–20 teeth; 20 for >20 teeth)

Ratchet Mechanism Types — Visual Reference

    Simple Ratchet           Reversible Ratchet        Frictional Ratchet

     ┌──╲                      ┌──╲                    ╭──────────╮
    ╱ ╲  ╲← Pawl              ╱ ╲  ╲← Double           │ ○ Roller │
   │ ○  ╲ ╲                  │ ○  ╲  ╲ Ended           │╱╲  ╱╲  ╱│
   │  ╲  ╲╱                  │  ╲  ╲╱╱ Pawl            ╰──────────╯
    ╲  ╲╱ Ratchet             ╲  ╲╱ Symmetric           Outer Ring
     ╲╱   Wheel                ╲╱  Teeth

Gear Tooth Anatomy: Sizes, Shapes, and Standards

How Tooth Size Is Specified

Gear teeth range from massive (in mining equipment and ship drives) to microscopic (in watch movements and instruments). The size is specified by one of two systems:

Diametral Pitch System (predominantly used in inch-based systems):

• P = Number of teeth per unit of pitch diameter
• Higher P = smaller teeth

Module System (used in metric systems):

• m = Pitch diameter (mm) ÷ Number of teeth
• Higher m = larger teeth

The Relationship

P × m = 25.4 (when m is in mm and P is in 1/inches)

Preferred Diametral Pitches

The industry uses standard pitch values to enable interchangeability:

Coarse Pitch (≤ 19P): 1, 1.25, 1.5, 1.75, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16

Fine Pitch (≥ 20P): 20, 24, 32, 48, 64, 72, 80, 96, 120, 150, 200

Standard Tooth Forms: ANSI B6.1-1968

The American National Standard establishes two involute spur gear forms:

1. 20-degree pressure angle — minimum 18 teeth (standard for nearly all modern gearing)
2. 25-degree pressure angle — minimum 12 teeth (greater strength, fewer teeth possible)

Both are defined by the basic rack — the theoretical rack with which all compliant gears are conjugate.

The Basic Rack

    ←── Circular Pitch (p) ──→

    ╱╲          ╱╲          ╱╲
   ╱  ╲        ╱  ╲        ╱  ╲      ← Addendum line
  ╱    ╲      ╱    ╲      ╱    ╲
 ╱──────╲────╱──────╲────╱──────╲─   ← Pitch line
 ╲      ╱    ╲      ╱    ╲      ╱
  ╲    ╱      ╲    ╱      ╲    ╱     ← Dedendum line
   ╲  ╱   r    ╲  ╱        ╲  ╱
    ╲╱ (fillet)  ╲╱          ╲╱

  │← t →│← space →│

  φ = Pressure Angle (20° or 25°)
  r = Fillet radius
  t = Tooth thickness at pitch line = p/2

Comparative Tooth Shapes

The appearance of gear teeth changes dramatically with pressure angle:

• 14.5° teeth: Tall, slender, deeper engagement — now obsolete for new designs
• 20° teeth: Balanced proportions, good strength, standard for modern gears
• 25° teeth: Shorter, wider, stronger — allows fewer teeth before undercut occurs

And with diametral pitch:

• P = 2: Large, robust teeth (half-inch pitch diameter per tooth)
• P = 10: Medium teeth (0.1-inch pitch diameter per tooth)
• P = 48: Fine teeth (barely visible to the eye)