Geometry Worksheets

Practice and master geometry concepts with helpful walkthroughs from our elite math educators, then try practice math problems to sharpen your skills.

Scale Factor

By Alison Rodriguez

A scale factor is a number (can be a whole number, decimal, or fraction) that is multiplied to another number to produce a larger or smaller number. It is often described as the ratio when comparing things like a blueprint, a map, or a scaled model of something. You can find it by creating this ratio between corresponding parts of the two figures.

Why is this concept useful?

Scale factors are often used in geometry to help dilate figures. However, the scale factor is used all the time in real life! Those wanting to create a model of a real-life object, such as a plane, a statue, or a building, can use the scale factor to give accurate measurements on a smaller scale. By using it, we can make sure that the model will have the same measurements, just in a smaller size. A cartographer uses a scale factor to draw a map in order for one to be able to hold it in their hands and still get an accurate representation of the land around them.

Where does this concept fit into the curriculum?

7th and 8th grade, High School Geometry

How can we use scale factor?

Scale factors help to visualize the large, real-world objects in a much smaller size. Typically, the larger measurement is the numerator and the smaller number is the denominator. If the scale factor is a whole number or the ratio is a fraction greater than 1, then the copy of it will be larger than the original. If the scale factor or ratio is a fraction between 0 and 1, then the copy will be smaller than the original. We can write scale factors as a ratio or as a fraction, just like ratios can!

Visualize it through the following model:

The original copy is in the middle. See how when you have a scale factor of 2:1, the copy gets larger. However, when the scale factor is ½, you can see the copy gets smaller. Take a look at the measurements on the sides of the rectangle. You can see the larger one’s dimensions are double and the smaller one’s dimensions are exactly half.

If you are trying to determine the scale factor between two objects, you can use the following formulas.

To scale up, smaller to larger, you can use:

LargerFigureMeasurementSmallerFigureMeasurement\frac{Larger Figure Measurement}{Smaller Figure Measurement}

To scale down, larger to smaller, you can use:

SmallerFigureMeasurementLargerFigureMeasurement\frac{Smaller Figure Measurement}{Larger Figure Measurement}

Sample Math Problems

Question

The images below are similar figures. Find the scale factor from the image on the left to the image on the right.

Answer

Since we are finding the scale factor of the image on the left to the image on the right, we want a scale factor that will be greater than one. This is because the image on the right is larger than the one on the left. We will need to use measurements from the corresponding sides. In this case, we use the top of both figures.

LargerFigureMeasurementSmallerFigureMeasurement=917=131\frac{LargerFigure Measurement}{Smaller Figure Measurement} = \frac{91}{7} = \frac{13}{1}

Question

Find the perimeter of the new image after being dilated with a scale factor of 7.

Answer

First we need to find the dimensions of the dilated figure. Since the scale factor is greater than 1, we know we are dilating to a larger image. We will multiply each of the side lengths by the scale factor.

18 × 7 = 126

This diagram is a square so each side will be 126 miles. To find the perimeter of a square we use the formula P = 4s.

P = 4 × 126 = 504

The perimeter of the new image would be 504 miles.

Question

Adeline wants to enlarge her photo to hang up in her bedroom. The current picture is 5 inches by 7 inches. She wants the new picture to be 4 times as large as the original. Find the scale factor that compares the areas of the smaller to the larger image.

Answer

The current photo has an area of 5 × 7 = 35 in². To find the new dimensions, we use the scale factor of 4:1. The new dimensions would be as follows:

5 × 4 = 20 in²

7 × 4 = 28 in²


The area of the enlarged photo is 20 × 28 = 560 in²

The scale factor of the areas of the smaller to the larger image is 35:560  or  1:16.

Question

George uses a mirror to look at the top of the school. He creates two similar triangles. Find the height of the building.

Answer

The distance from the building to the mirror is found by subtracting, 181.5 – 1.5 = 180 m

The scale factor of the smaller triangle to the larger one is

180:1.5

1800:15   (multiply by 10 to get rid of the decimal)

120:1

Therefore, the height of the school is found by multiplying 1.25 x 120 = 150.

The building is 150 meters tall.

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Practice Math Problems

1. The images below are similar figures. Find the scale factor ∆CAT to ∆DOG.

2.  Find the perimeter of the image after the dilation.

3. The distance from Philadelphia to New York City is marked on a map as being 13.5 inches apart. The scale reads 1 inch : 6  miles.  How far apart are Philadelphia and New York City?

4. Michael is 6 feet tall and is standing outside next to his younger sister. His shadow is 8 feet long. His sister’s shadow is 5 feet long. How tall is his sister?

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