{
  "id": "earth-curvature",
  "version": "836fed07e2d7",
  "status": "published",
  "name": "Earth Curvature Calculator",
  "question": "What does the earth curvature hide?",
  "summary": "Works out how far the Earth’s surface curves away below a level line over a distance, the distance to the horizon from a height, and how much of a far object the curvature hides.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/earth-curvature-calculator",
  "markdown": "https://www.acalculator.org/physics/earth-curvature-calculator.md",
  "kind": "function",
  "method": "θ = d ÷ R; drop = R(sec θ − 1); horizon = √(h(2R + h)); θh = atan(horizon ÷ R); hidden = R(sec(θ − θh) − 1) beyond the horizon, else 0.",
  "assumptions": [
    "The Earth is a smooth sphere of radius 6,371 km (NASA’s mean radius); the distance is measured along its surface.",
    "Light travels in straight lines: no atmospheric refraction, which in practice bends sight lines down and hides less."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "d": {
        "title": "Distance",
        "description": "How far away the object is, measured along the surface.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.01,
        "maximum": 5000000
      },
      "h": {
        "title": "Eye height",
        "description": "How high your eyes (or the camera) are above the surface: 0 for the plain curvature drop.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0,
        "maximum": 100000
      },
      "R": {
        "title": "Earth radius",
        "description": "The radius of the sphere: 6,371 km is Earth’s mean radius.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1000,
        "maximum": 100000000
      }
    }
  },
  "outputs": {
    "hidden": {
      "label": "Hidden height",
      "description": "How much of a far object is below your horizon: 0 if the object’s base is in view.",
      "format": "quantity",
      "unit": "ft"
    },
    "drop": {
      "label": "Curvature drop",
      "description": "How far the surface at that distance is below a level line from your feet: R(sec θ − 1).",
      "format": "quantity",
      "unit": "ft"
    },
    "horizon": {
      "label": "Distance to the horizon",
      "description": "The straight-line distance from your eyes to the horizon: √(h(2R + h)).",
      "format": "quantity",
      "unit": "mi"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "d": "10 mi",
      "h": "6 ft",
      "R": "6371 km"
    },
    "outputs": {
      "hidden": 9.961284384629236,
      "drop": 20.32644009093313,
      "horizon": 4827.273862596719
    },
    "text": "At 10 mi, the curve hides 32.6814 ft of a far object; the horizon is 2.99953 mi away."
  },
  "examples": [
    {
      "given": {
        "d": 1609.344,
        "h": 0,
        "R": 6371000
      },
      "expect": {
        "drop": 0.20326386589203657,
        "hidden": 0.20326386589203657,
        "horizon": 0
      },
      "source": "NASA, Earth Fact Sheet (volumetric mean radius 6,371.000 km), https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html; hand calculation in content.mdx: θ = 1,609.344 ÷ 6,371,000; drop = R(sec θ − 1) = 0.203264 m = 8.00251 in"
    },
    {
      "given": {
        "d": 16093.44,
        "h": 1.8288,
        "R": 6371000
      },
      "expect": {
        "hidden": 9.961284384629236,
        "horizon": 4827.273862596719
      },
      "source": "Wikipedia, Horizon: Distance to the horizon (d = √(2Rh + h²) for an observer at height h on a sphere of radius R), https://en.wikipedia.org/wiki/Horizon; NASA, Earth Fact Sheet (volumetric mean radius 6,371.000 km), https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html; hand calculation in content.mdx: horizon √(1.8288 × (2 × 6,371,000 + 1.8288)) = 4,827.27 m; hidden 9.96128 m = 32.6814 ft"
    },
    {
      "given": {
        "d": 3000,
        "h": 2,
        "R": 6371000
      },
      "expect": {
        "hidden": 0,
        "horizon": 5048.168380709978
      },
      "source": "Wikipedia, Horizon: Distance to the horizon (d = √(2Rh + h²) for an observer at height h on a sphere of radius R), https://en.wikipedia.org/wiki/Horizon; hand calculation in content.mdx: horizon √(2 × 12,742,002) = 5,048.17 m, more than 3,000 m"
    }
  ],
  "sources": [
    "NASA, Earth Fact Sheet (volumetric mean radius 6,371.000 km). https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html (retrieved 2026-10-03)",
    "Wikipedia, Horizon: Distance to the horizon (geometric distance d = √(2Rh + h²)). https://en.wikipedia.org/wiki/Horizon (retrieved 2026-10-03)",
    "NIST SP 811, Appendix B.8 (1 mi = 1,609.344 m; 1 ft = 0.3048 m; 1 nautical mile = 1,852 m). https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-03)"
  ],
  "related": [
    "field-of-view",
    "speed",
    "free-fall"
  ],
  "changelog": []
}