A scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.
Every length is multiplied by the same scale factor, here 2. The area is multiplied by 22 = 4, because both the base and the height doubled.
Formal definition and notation
If figure B is a scaled copy of figure A, every length in B equals k times the matching length in A; the positive constant k is the scale factor from A to B, k = image lengthoriginal length. If k > 1 the copy is an enlargement, if 0 < k < 1 it is a reduction, and if k = 1 it is the same size. Angles do not change, so the figures are similar. Perimeters are multiplied by k, areas by k2 and volumes by k3.
a drawing scale: 1 unit on the drawing stands for 50 of the same units in real life (scale factor 150)
Key facts
k > 1 enlarges, 0 < k < 1 reduces, and k = 1 gives a congruent copy.
The scale factor from the copy back to the original is 1k: enlarging by 4 is undone by scaling by 14.
Map and blueprint scales are scale factors: on a 1 : 50,000 map, 1 cm stands for 50,000 cm, which is 0.5 km.
A scale factor of 3 multiplies perimeter by 3, area by 9 and volume by 27.
A dilation centered at the origin multiplies both coordinates by k: with k = 3, (2, −1) moves to (6, −3).
Finding and using a scale factor
Finding and using a scale factor
k = new lengthoriginal length
new length
a side of the scaled copy (the image)
original length
the matching side of the original figure
k
new length = k × original length
Divide new by original, not the other way round. Then perimeter changes by k, area by k2 and volume by k3.
Try it: dilate a shape
Interactive
Choose a scale factor of one half, 2 or 3. The readout lists each original point and its image; compare the side lengths of the blue original and the green image.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Scale Factor examples, step by step
Example 1: Enlarge a rectangle
Warm-up
A rectangle is 4 cm by 7 cm. It is enlarged with a scale factor of 3. What are the new dimensions?
Multiply every length by the scale factor 3.
4 × 3 = 12 cm.
7 × 3 = 21 cm.
Answer: The new rectangle is 12 cm by 21 cm.
Example 2: Find the scale factor of a reduction
Core
A triangle with sides 10, 14 and 18 is reduced to a similar triangle whose shortest side is 4. Find the scale factor and the other two sides.
Match the shortest sides: k = 410 = 25.
k is less than 1, which fits a reduction.
14 × 25 = 5.6 and 18 × 25 = 7.2.
Answer: The scale factor is 2/5 (0.4), so the other sides are 5.6 and 7.2.
Example 3: What happens to the area
Test-style
A 6 in by 4 in photo is enlarged with a scale factor of 2.5. How many times larger is the area, and what is the new area?
A dilation is the transformation that resizes a figure from a center point. The scale factor is the number that tells the dilation how much to stretch or shrink.
A scale factor is a ratio of matching lengths, written as one number. A drawing scale such as 1 : 50 is the same idea written as a ratio.
Quick check: 3 questions
Question 1 of 3
A segment of length 9 is dilated with a scale factor of 23. What is the length of the image?
Show answer and explanation
Answer: A.9 × 23 = 6. Choice B divides by 23 instead of multiplying, C adds 2, and D multiplies by 13 only.
Question 2 of 3
A 2 by 5 rectangle is scaled to a similar 8 by 20 rectangle. What is the scale factor?
Show answer and explanation
Answer: B.k = 82 = 205 = 4. Choice C divides original by new, D is the factor for the area (42), and A is just a difference of lengths.
Question 3 of 3
A square with side 3 cm is scaled by a factor of 4. What is the area of the new square?
Show answer and explanation
Answer: C. The new side is 3 × 4 = 12 cm, so the area is 122 = 144 square cm (that is 9 × 42). Choice A multiplies the area by 4 instead of 16, B is the new perimeter, and D is the new side.
Scale Factor: frequently asked questions
How do you find the scale factor?
Pick a pair of matching sides, one from the original and one from the copy, and divide: new length over original length. If a 5 cm side becomes 20 cm, the scale factor is 205 = 4. Check with a second pair of sides; in similar figures every pair gives the same number.
What does a scale factor less than 1 mean?
It means the copy is smaller than the original, a reduction. A scale factor of 12 halves every length, and 0.1 makes each length one tenth as long. Maps and blueprints use very small scale factors, such as 1100, so that large real objects fit on a page.
How does the scale factor change area and volume?
Area changes by the square of the scale factor and volume by the cube. With k = 2, a figure’s area becomes 4 times as large, and a solid’s volume becomes 8 times as large. That is because area uses two lengths (length times width) and volume uses three.
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