Scale Factors
The scale factor is the ratio of a new figure's dimensions to the corresponding dimensions of the original figure. To find new dimensions, multiply the original by the scale factor: new length = scale factor x original length. For example, if a map scale factor is 1/10,000, a 3 cm line on the map represents 30,000 cm or 300 meters in reality. This Grade 8 math skill from Yoshiwara Core Math Chapter 6 is essential for working with maps, architectural plans, scale models, and similar figures. Scale factors connect ratio and proportion directly to practical geometric applications.
Key Concepts
Property The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure. $$ \text{scale factor} = \frac{\text{new length}}{\text{corresponding original length}} $$ $$ \text{new length} = \text{scale factor} \times \text{corresponding original length} $$.
Examples A map has a scale factor of $\frac{1}{10000}$. A road that is 3 cm long on the map is $3 \times 10000 = 30000$ cm, or 300 meters, in real life. A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is $\frac{20}{8} = 2.5$. To reduce a 12 foot wall to fit on a blueprint with a scale factor of $\frac{1}{48}$, its length on the blueprint is $12 \times \frac{1}{48} = \frac{1}{4}$ foot, or 3 inches.
Explanation The scale factor is the number you multiply by to change the size of a figure. A scale factor greater than 1 makes the figure bigger (an enlargement), while a factor between 0 and 1 makes it smaller (a reduction).
Common Questions
What is a scale factor?
A scale factor is the ratio of a dimension in a new (scaled) figure to the corresponding dimension in the original figure. Scale factor = new length / original length. It tells you how much larger or smaller the new figure is.
How do you use a scale factor to find new dimensions?
Multiply the original dimension by the scale factor to get the new dimension. For example, if the original is 12 feet and the scale factor is 1/48, the scaled length is 12 x 1/48 = 1/4 foot = 3 inches.
What does a scale factor greater than 1 mean?
A scale factor greater than 1 means the new figure is larger than the original (an enlargement). For example, a scale factor of 2.5 means every dimension of the new figure is 2.5 times larger than the original.
When do 8th graders learn about scale factors?
Students study scale factors in Grade 8 math as part of Chapter 6 of Yoshiwara Core Math, which covers core concepts including similarity and scaling.
How is scale factor used in maps?
Map scale factors tell you the relationship between map distances and real distances. A scale factor of 1/10,000 means 1 cm on the map equals 10,000 cm (100 meters) in reality. This lets you calculate actual distances from maps.
What is the scale factor of a triangle with sides 8 inches enlarged to sides of 20 inches?
Scale factor = new length / original length = 20 / 8 = 2.5. Every dimension of the new triangle is 2.5 times the corresponding dimension of the original.