Scale Factor Calculator
Calculate scale factor (k) for dilations, enlargements, and scale drawings.
Scale Factor Formulas
Find k: k = New Size / Original Size
Apply k: New Size = Original Size × k
Area Multiplier: k²
Volume Multiplier: k³
Scale Factor Types & Applications
| Scale Factor | Type | Example Application |
|---|---|---|
| k = 0.5 | Reduction (50%) | Shrink photo to half size |
| k = 1 | Congruent (same) | No change in size |
| k = 2 | Enlargement (200%) | Double size (blueprint scale) |
| k = 1/100 | Reduction (1:100) | Map scale: 1 cm = 100 km |
Frequently Asked Questions
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A scale factor (k) is a multiplier that describes how much a figure is enlarged or reduced. If k > 1, the figure enlarges. If 0 < k < 1, the figure reduces. If k = 1, the figure stays the same size (congruent).
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Divide the new dimension by the original dimension: k = new size / original size. Example: If original = 5 cm and new = 15 cm, then k = 15/5 = 3 (three times larger).
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Dilation is a transformation that enlarges or reduces a figure from a center point. All distances from the center are multiplied by the scale factor. The shape stays the same; only the size changes.
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Enlargement: k > 1 (figure gets bigger). Example: k = 2 means double the size. Reduction: 0 < k < 1 (figure gets smaller). Example: k = 0.5 means half the size.
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If linear scale factor = k, then area scales by k². Example: If k = 2, areas multiply by 2² = 4. A shape with area 10 cm² dilated by k = 2 becomes 40 cm².
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If linear scale factor = k, then volume scales by k³. Example: If k = 2, volumes multiply by 2³ = 8. A shape with volume 10 cm³ dilated by k = 2 becomes 80 cm³.
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In standard dilation, scale factor is positive. Negative scale factors cause reflection and dilation (reversal + size change). Most geometry courses work with positive scale factors only.
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Scale notation like "1:100" means 1 unit on the drawing = 100 units in reality. This is equivalent to scale factor k = 1/100. Blueprints use similar ratios.
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Multiply every length, width, height by the scale factor. For a rectangle with dimensions 4 × 6 scaled by k = 2, new dimensions = 8 × 12.
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Figures with the same shape but different sizes. They have equal corresponding angles and sides in proportion (scaled by same factor k). Scale factor relates all corresponding sides.
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Maps, blueprints, model buildings, scale drawings, photo resizing, architectural plans. Any situation where a design is enlarged or reduced maintains scale factor relationships.
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The fixed point from which all distances are multiplied by the scale factor. Usually the origin (0,0) in coordinate geometry. Changing the center changes where the dilated figure ends up.