Key takeaway: A scale factor is the single multiplier applied to every length of a figure to build an enlarged or reduced copy that keeps the same shape.
Grade 77.G.A.1Geometry and Measurement
Definition of Scale Factor
Every pair of similar figures is tied together by one scale factor. When it exceeds 1 the copy grows. When it sits between 0 and 1 the copy shrinks. A factor of exactly 1 leaves the figure the same size.
Each pair of matching sides yields this same value, so comparing a few pairs shows whether two shapes are truly scaled copies of one another. A scale factor of 1 means a perfect duplicate. Matching angles stay equal. Photo prints, model trains and zoomed screen images are everyday cases.
Formula and notation
Put an image side over its partner side in the original:
\[k = \frac{\text{image length}}{\text{original length}}\]
Lengths are multiplied by \(k\), areas by \(k^2\), and volumes by \(k^3\).
Key facts
- A value above 1 enlarges, a value below 1 shrinks, and exactly 1 changes nothing.
- The factor from the image back to the original is the reciprocal, \(\frac{1}{k}\).
- Areas grow by the square of the multiplier, so doubling lengths quadruples area.
- Corresponding angles are unchanged, only the lengths are rescaled.
Scale Factor example
Lucia prints a snapshot that is 4 inches wide and 6 inches tall as a poster 10 inches wide and 15 inches tall. Find the multiplier for the poster, then the poster height of a 5-inch fence post pictured in the snapshot.
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Compare the matching sides:\(\frac{15}{6} = 2.5\) and \(\frac{10}{4} = 2.5\).
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Both agree, so \(k = 2.5\) and the poster is an enlargement.
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Fence post on the poster: \(5 \times 2.5 = 12.5\) inches.
The factor is 2.5, and the fence post stands 12.5 inches tall on the poster.
Common mistake
Watch out
Ravi says the poster covers 2.5 times the area of the snapshot, so it is \(24 \times 2.5 = 60\) square inches.
Why it is wrong
The multiplier applies to lengths. Area involves two lengths, so it grows by \(2.5^2 = 6.25\). The poster’s area is \(24 \times 6.25 = 150\) square inches, matching \(10 \times 15 = 150\).
Related terms and lessons
- Grade 7 · Scale DrawingsLessonWorksheetQuiz
- Grade 8 · Congruence, Dilations and SimilarityLessonWorksheetQuiz
- Geometry · Dilations and Similarity TransformationsLessonWorksheetQuiz
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