{
  "id": "scale-factor",
  "version": "689d25cf69d1",
  "status": "published",
  "name": "Scale Factor Calculator",
  "question": "What is the scale factor?",
  "summary": "Computes the scale factor between two similar figures from an original and a new length (k = new ÷ original), or a length from the factor, with the area factor k², the volume factor k³ and the scale ratio.",
  "category": "math",
  "subcategory": "geometry",
  "url": "https://www.acalculator.org/math/scale-factor-calculator",
  "markdown": "https://www.acalculator.org/math/scale-factor-calculator.md",
  "kind": "formulas",
  "method": "k = b ÷ a; b = k × a; a = b ÷ k; area factor = k²; volume factor = k³.",
  "assumptions": [
    "The two figures are similar: every length is scaled by the same factor.",
    "Both lengths are in the same unit, and are more than 0."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "a": {
        "title": "Original length",
        "description": "A length on the original figure, in any unit.",
        "type": "number",
        "exclusiveMinimum": 0
      },
      "b": {
        "title": "New length",
        "description": "The matching length on the new (scaled) figure, in the same unit.",
        "type": "number",
        "exclusiveMinimum": 0
      },
      "k": {
        "title": "Scale factor",
        "description": "How many times larger the new figure is: new length ÷ original length.",
        "type": "number",
        "minimum": 1e-100,
        "maximum": 1e+100
      }
    }
  },
  "solve": {
    "fillAny": 2,
    "variables": [
      "a",
      "b",
      "k"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "a": {
      "label": "Original length",
      "description": "A length on the original figure, in any unit.",
      "format": "number"
    },
    "b": {
      "label": "New length",
      "description": "The matching length on the new (scaled) figure, in the same unit.",
      "format": "number"
    },
    "k": {
      "label": "Scale factor",
      "description": "New length ÷ original length.",
      "format": "number"
    },
    "fraction": {
      "label": "Scale factor as a fraction",
      "description": "The factor as a fraction in lowest terms, from the two lengths as typed (shown when its bottom is at most 1,000,000).",
      "format": "text"
    },
    "ratio": {
      "label": "Scale ratio",
      "description": "The scale as new : original, written 1 : n for a reduction and k : 1 for an enlargement.",
      "format": "text"
    },
    "area": {
      "label": "Area scale factor",
      "description": "How many times larger every area is: k².",
      "format": "number"
    },
    "volume": {
      "label": "Volume scale factor",
      "description": "How many times larger every volume is: k³.",
      "format": "number"
    },
    "kind": {
      "label": "Type of scaling",
      "description": "An enlargement when k > 1, a reduction when k < 1, the same size when k = 1.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "a": 4,
      "b": 10
    },
    "outputs": {
      "k": 2.5,
      "fraction": "2 1/2",
      "ratio": "2.5 : 1",
      "area": 6.25,
      "volume": 15.625,
      "kind": "Enlargement"
    },
    "text": "From 4 to 10, the scale factor is 2.5 (2.5 : 1)."
  },
  "examples": [
    {
      "given": {
        "a": 4,
        "b": 10
      },
      "expect": {
        "k": 2.5,
        "area": 6.25,
        "volume": 15.625,
        "ratio": "2.5 : 1",
        "kind": "Enlargement"
      },
      "source": "OpenStax Elementary Algebra 2e, section 8.7, Solve Proportion and Similar Figure Applications: corresponding sides of similar figures are in the same ratio (https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications, retrieved 2026-10-01); hand calculation in content.mdx: 10 ÷ 4 = 2.5, 2.5² = 6.25, 2.5³ = 15.625"
    },
    {
      "given": {
        "a": 1.3,
        "k": 270
      },
      "expect": {
        "b": 351
      },
      "source": "OpenStax Elementary Algebra 2e, section 8.7, Example 8.79: 1 inch on the map is 270 miles, 1.3 inches is 351 miles (https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "a": 9,
        "b": 3
      },
      "expect": {
        "k": 0.3333333333333333,
        "ratio": "1 : 3",
        "kind": "Reduction",
        "volume": 0.037037037037037035
      },
      "source": "OpenStax Elementary Algebra 2e, section 8.7 (similar figures); hand calculation in content.mdx: 3 ÷ 9 = 1/3, (1/3)³ = 1/27"
    }
  ],
  "sources": [
    "OpenStax, Elementary Algebra 2e, section 8.7, Solve Proportion and Similar Figure Applications (similar figures have corresponding sides in the same ratio; Example 8.79, map scale), CC BY 4.0, retrieved 2026-10-01. https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications"
  ],
  "related": [
    "ratio",
    "proportion"
  ],
  "changelog": []
}