Scale Factor
In short: Scale factor names the single number you multiply every length of a figure by to get the matching length in an enlarged or reduced copy of that figure.
7th GradeGeometry7.G.A.1
What is Scale Factor?
The scale factor of two similar figures is the constant multiplier linking one to the other. Blow a photo up from 4 by 6 inches to 10 by 15 inches and each side gets multiplied by the same number, 2.5.
Find it as a ratio of matching sides: \(k = \frac{\text{new length}}{\text{original length}}\). A value above 1 enlarges, a value between 0 and 1 shrinks, and exactly 1 leaves the size alone. Angles never change; only lengths do, and areas grow by \(k^2\).
How to spot it
- A shape is copied larger or smaller and the problem asks how many times as long the new sides are.
- Two figures look alike, with equal angles, but their matching sides have different lengths.
- A photo, logo or poster is resized and you must find the new width from the old height.
- A model is described as one-twelfth the size of the real thing.
The method
- Pick one pair of matching sides, one from the original and one from the copy.
- Divide the copy’s length by the original’s length to obtain \(k\).
- Check a second pair of matching sides to confirm that the quotient is the same.
- Multiply any original length by \(k\) to predict its partner, or divide a copy length by \(k\) to go back.
Scale Factor: solved example
Problem
A print shop enlarges a 4-inch by 6-inch photo into a poster 10 inches tall. How wide is the poster, and what is the scale factor?
Solution
The photo is 6 inches wide and 4 inches tall; the poster is 10 inches tall. The factor is \(k = \frac{10}{4} = 2.5\).
The new width is \(6 \times 2.5 = 15\) inches. The scale factor is 2.5 and the poster is 15 inches wide.
Try it
Question 1
A banner design is 36 inches long; the pocket-card version is 9 inches long. What is the scale factor from banner to card?
Show the answer
\(\frac{9}{36} = \frac{1}{4}\), a reduction.
Question 2
A triangle has a side of 12 cm. After a scale factor of 0.75, how long is that side?
Show the answer
\(12 \times 0.75 = 9\) cm.
Scale Factor vs. Scale Drawing
The scale factor is one number, the multiplier between matching lengths of two figures. A scale drawing is the whole picture or plan produced using that number, and its scale is written as a ratio with units, like 1 inch to 8 feet. First comes the picture with its stated scale; the factor is what you compute from it.
Keep going
Lessons
Worksheets and quizzes
- Scale Drawings – Worksheet
- Scale Drawings – Quiz
- Dilations and Similar Figures – Worksheet
- Dilations and Similar Figures – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Geometry