{
  "id": "contact-lens-vertex",
  "version": "95efb28b4a55",
  "status": "published",
  "name": "Contact Lens Vertex Calculator",
  "question": "What is my contact lens vertex?",
  "summary": "Converts a glasses prescription to contact lens power at the eye (or back) with the vertex formula Fc = F ÷ (1 − d·F), for the sphere and the cylinder, rounded to the 0.25 D steps lenses come in.",
  "category": "health",
  "subcategory": "body",
  "url": "https://www.acalculator.org/health/contact-lens-vertex-calculator",
  "markdown": "https://www.acalculator.org/health/contact-lens-vertex-calculator.md",
  "kind": "function",
  "method": "Glasses to contacts: Fc = F ÷ (1 − d × F); contacts to glasses: F = Fc ÷ (1 + d × Fc); d is the vertex distance in metres. The cylinder converts both meridians (S and S + C) and subtracts.",
  "assumptions": [
    "A soft contact lens sits on the cornea, so its vertex distance is 0.",
    "The cylinder axis does not change.",
    "Contact lenses come in 0.25 D steps here; strong powers are often made in 0.50 D steps.",
    "This is an optical conversion, not a fitting; an eye care professional fits and prescribes contact lenses."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "from": {
        "title": "Convert from",
        "description": "Convert a glasses prescription to contact lenses, or contact lenses to glasses.",
        "type": "string",
        "enum": [
          "glasses",
          "contacts"
        ]
      },
      "sph": {
        "title": "Sphere (D)",
        "description": "The sphere power in diopters, minus for nearsighted and plus for farsighted.",
        "type": "number",
        "minimum": -25,
        "maximum": 25
      },
      "cyl": {
        "title": "Cylinder (D)",
        "description": "Optional: the cylinder power for astigmatism, in diopters, with its sign.",
        "type": "number",
        "minimum": -10,
        "maximum": 10
      },
      "vd": {
        "title": "Vertex distance (mm)",
        "description": "The distance from the back of the glasses lens to the front of the eye; 12 mm is usual.",
        "type": "number",
        "minimum": 5,
        "maximum": 20
      }
    }
  },
  "outputs": {
    "sphere": {
      "label": "Sphere at the new plane",
      "description": "The sphere power after the vertex change, to the nearest 0.25 D.",
      "format": "number"
    },
    "cylinder": {
      "label": "Cylinder at the new plane",
      "description": "The cylinder power after the vertex change, to the nearest 0.25 D.",
      "format": "number"
    },
    "sphereExact": {
      "label": "Sphere, unrounded",
      "description": "The sphere after the vertex change before rounding to 0.25 D.",
      "format": "number"
    },
    "cylinderExact": {
      "label": "Cylinder, unrounded",
      "description": "The cylinder after the vertex change before rounding to 0.25 D.",
      "format": "number"
    },
    "matters": {
      "label": "Does vertex matter?",
      "description": "Vertex changes the power noticeably when a meridian is stronger than ±4.00 D.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "from": "glasses",
      "sph": -6,
      "vd": 12
    },
    "outputs": {
      "sphere": -5.5,
      "sphereExact": -5.597014925373134,
      "matters": "Yes: the power is over ±4.00 D"
    },
    "text": "A -5.5 D sphere at the new plane."
  },
  "examples": [
    {
      "given": {
        "from": "glasses",
        "sph": -6,
        "vd": 12
      },
      "expect": {
        "sph": -5.5,
        "sphereExact": -5.597014925373134
      },
      "source": "Wikipedia, Vertex distance: Fc = F ÷ (1 − xF) with x the vertex change in metres (positive toward the eye); 12 to 14 mm is usual; it matters beyond about ±4.00 D, https://en.wikipedia.org/wiki/Vertex_distance (retrieved 2026-10-05); hand calculation in content.mdx: −6 ÷ (1 + 0.072) = −5.597, nearest 0.25 D step −5.50"
    },
    {
      "given": {
        "from": "glasses",
        "sph": 8,
        "vd": 12
      },
      "expect": {
        "sph": 8.75,
        "sphereExact": 8.849557522123893
      },
      "source": "Wikipedia, Vertex distance: Fc = F ÷ (1 − xF) with x the vertex change in metres (positive toward the eye); 12 to 14 mm is usual; it matters beyond about ±4.00 D, https://en.wikipedia.org/wiki/Vertex_distance (retrieved 2026-10-05); hand calculation in content.mdx: 8 ÷ (1 − 0.096) = 8.850, nearest step 8.75"
    },
    {
      "given": {
        "from": "glasses",
        "sph": -5,
        "cyl": -2,
        "vd": 12
      },
      "expect": {
        "sph": -4.75,
        "cyl": -1.75,
        "sphereExact": -4.716981132075471,
        "cylinderExact": -1.7405834435702845
      },
      "source": "Wikipedia, Vertex distance: Fc = F ÷ (1 − xF) with x the vertex change in metres (positive toward the eye); 12 to 14 mm is usual; it matters beyond about ±4.00 D, https://en.wikipedia.org/wiki/Vertex_distance (retrieved 2026-10-05); hand calculation in content.mdx: −5 ÷ 1.06 = −4.717 (−4.75); −7 ÷ 1.084 = −6.458 (−6.50); cylinder −6.50 − (−4.75) = −1.75"
    },
    {
      "given": {
        "from": "contacts",
        "sph": -5.5,
        "vd": 12
      },
      "expect": {
        "sph": -6,
        "sphereExact": -5.8886509635974305
      },
      "source": "Wikipedia, Vertex distance: Fc = F ÷ (1 − xF) with x the vertex change in metres (positive toward the eye); 12 to 14 mm is usual; it matters beyond about ±4.00 D, https://en.wikipedia.org/wiki/Vertex_distance (retrieved 2026-10-05); hand calculation in content.mdx: −5.5 ÷ (1 − 0.066) = −5.889, nearest step −6.00"
    }
  ],
  "sources": [
    "Wikipedia. Vertex distance: the effective power formula Fc = F ÷ (1 − xF), 12 to 14 mm usual, significant beyond about ±4.00 D, retrieved 2026-10-05. https://en.wikipedia.org/wiki/Vertex_distance",
    "Federal Trade Commission. Buying Prescription Glasses or Contact Lenses: Your Rights (sellers may sell contact lenses only with a valid prescription), retrieved 2026-10-05. https://consumer.ftc.gov/articles/buying-prescription-glasses-or-contact-lenses-your-rights"
  ],
  "related": [],
  "changelog": []
}