{
  "id": "gear-ratio",
  "version": "fc6f49112ed8",
  "status": "published",
  "name": "Gear Ratio Calculator",
  "question": "What is the gear ratio?",
  "summary": "Computes the gear ratio of a driving and a driven gear from their teeth, and the output speed in rpm from the input speed, or a missing teeth count or speed from the other three.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/gear-ratio-calculator",
  "markdown": "https://www.acalculator.org/physics/gear-ratio-calculator.md",
  "kind": "formulas",
  "method": "z₁ × n₁ = z₂ × n₂ (teeth × rpm is the same on both gears); ratio = z₂ ÷ z₁; n₂ = n₁ ÷ ratio.",
  "assumptions": [
    "Two gears (or two sprockets on a chain) mesh directly. For a train of several pairs, multiply the ratios of the pairs.",
    "Tooth counts are from 1 to 10,000 and speeds up to 10,000,000 rpm. A solved value outside that range has no answer.",
    "The teeth do not slip, so the teeth that pass the contact point per minute are the same on both gears.",
    "The ideal output torque is the input torque times the ratio. Real gears lose a few percent to friction; this calculator does not include losses."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "z1": {
        "title": "Driving gear teeth",
        "description": "The number of teeth on the driving (input) gear.",
        "type": "number",
        "minimum": 1,
        "maximum": 10000
      },
      "z2": {
        "title": "Driven gear teeth",
        "description": "The number of teeth on the driven (output) gear.",
        "type": "number",
        "minimum": 1,
        "maximum": 10000
      },
      "n1": {
        "title": "Input speed (rpm)",
        "description": "The speed of the driving gear, in revolutions per minute.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 10000000
      },
      "n2": {
        "title": "Output speed (rpm)",
        "description": "The speed of the driven gear, in revolutions per minute.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 10000000
      }
    }
  },
  "solve": {
    "fillAny": 3,
    "variables": [
      "z1",
      "z2",
      "n1",
      "n2"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "driver": {
      "label": "Driving gear teeth",
      "description": "The number of teeth on the driving (input) gear.",
      "format": "number"
    },
    "driven": {
      "label": "Driven gear teeth",
      "description": "The number of teeth on the driven (output) gear.",
      "format": "number"
    },
    "inputRpm": {
      "label": "Input speed (rpm)",
      "description": "The speed of the driving gear, in revolutions per minute.",
      "format": "number"
    },
    "outputRpm": {
      "label": "Output speed (rpm)",
      "description": "The speed of the driven gear, in revolutions per minute.",
      "format": "number"
    },
    "ratio": {
      "label": "Gear ratio",
      "description": "Driven teeth divided by driving teeth: how many turns of the input make one turn of the output.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "z1": 12,
      "z2": 36,
      "n1": 1800
    },
    "outputs": {
      "outputRpm": 600,
      "ratio": 3
    },
    "text": "A 12-tooth gear driving a 36-tooth gear is a ratio of 3, so 1,800 rpm in gives 600 rpm out."
  },
  "examples": [
    {
      "given": {
        "z1": 12,
        "z2": 36,
        "n1": 1800
      },
      "expect": {
        "ratio": 3,
        "n2": 600
      },
      "source": "hand calculation in content.mdx: ratio = 36 ÷ 12 = 3 (3:1); output = 1,800 ÷ 3 = 600 rpm"
    },
    {
      "given": {
        "z1": 50,
        "z2": 20,
        "n1": 90
      },
      "expect": {
        "ratio": 0.4,
        "n2": 225
      },
      "source": "hand calculation in content.mdx: ratio = 20 ÷ 50 = 0.4; the wheel turns at 90 ÷ 0.4 = 225 rpm",
      "tolerance": 1e-12
    },
    {
      "given": {
        "z1": 15,
        "n1": 3000,
        "n2": 750
      },
      "expect": {
        "z2": 60,
        "ratio": 4
      },
      "source": "hand calculation in content.mdx: driven teeth = 15 × 3,000 ÷ 750 = 60; ratio = 60 ÷ 15 = 4"
    }
  ],
  "sources": [
    "ISO 1122-1:1998, Vocabulary of gear terms, Part 1: Definitions related to geometry. Gear ratio u = z₂ ÷ z₁ and transmission ratio i = ω₁ ÷ ω₂. https://www.iso.org/standard/5725.html",
    "Budynas, R. G., and J. K. Nisbett. Shigley's Mechanical Engineering Design, 11th edition, McGraw-Hill (2020), chapter 13, Gears: general, section 13-12 (gear trains)."
  ],
  "related": [
    "compression-ratio",
    "speed"
  ],
  "changelog": []
}