how to solve scale factor problems

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Education and Communications
Mathematics

How to Find Scale Factor

Last Updated: September 15, 2022 References

This article was co-authored by Mario Banuelos, PhD . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. This article has been viewed 820,442 times.

The scale factor, or linear scale factor, is the ratio of two corresponding side lengths of similar figures. Similar figures have the same shape but are of different sizes. The scale factor is used to solve geometric problems. [1] X Research source You can use the scale factor to find the missing side lengths of a figure. Conversely, you can use the side lengths of two similar figures to calculate the scale factor. These problems involve multiplication or require you to simplify fractions.

Finding the Scale Factor of Similar Figures

The problem should tell you that the shapes are similar, or it might show you that the angles are the same, and otherwise indicate that the side lengths are proportional, to scale, or that they correspond to each other.

For example, you might have a triangle with a base that is 15 cm long, and a similar triangle with a base that is 10 cm long.

Finding a Similar Figure Using the Scale Factor

For example, you might have a right triangle with sides measuring 4 cm and 3 cm, and a hypotenuse 5 cm long.

For example, if the scale factor is 2, then you are scaling up, and a similar figure will be larger than the one you have.

Completing Sample Problems

Irregular figures can be similar if all of their sides are in proportion. Thus, you can calculate a scale factor using any dimension you are given. [10] X Research source

${\text{Scale Factor}}={\frac {14}{8}}$

Finding the Scale Factor in Chemistry

H2O * 3 = H6O3

For example, the scaling factor for the compound is 3. The molecular formula of the compound is H6O3.

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Video . By using this service, some information may be shared with YouTube.

About This Article

To find scale factor, start by finding the length of a corresponding side on each figure. If you're scaling up from a smaller figure to a larger one, plug the lengths into the equation scale factor = larger length over smaller length. If you're scaling down from a larger figure to a smaller one, use the equation scale factor = smaller length over larger length. Plug in the lengths and simplify the fraction to find the scale factor. If you want to learn how to find the scale factor in chemistry, keep reading the article! Did this summary help you? Yes No

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Math Article

Scale Factor

Scale Factor is used to scale shapes in different dimensions . In geometry, we learn about different geometrical shapes which both in two-dimension and three-dimension. The scale factor is a measure for similar figures , who look the same but have different scales or measures. Suppose, two circle looks similar but they could have varying radii.

The scale factor states the scale by which a figure is bigger or smaller than the original figure. It is possible to draw the enlarged shape or reduced shape of any original shape with the help of scale factor.

Table of contents:

Scale Factor of Enlargement

Scale Factor of Triangle

Real-Life Applications

What is the Scale factor

The size by which the shape is enlarged or reduced is called as its scale factor. It is used when we need to increase the size of a 2D shape , such as circle, triangle, square, rectangle, etc.

If y = Kx is an equation, then K is the scale factor for x. We can represent this expression in terms of proportionality also:

Hence, we can consider K as a constant of proportionality here.

The scale factor can also be better understood by Basic Proportionality Theorem .

Scale Factor Formula

The formula for scale factor is given by:

Dimensions of Original Shape x scale Factor = Dimension of new shape

Scale factor = Dimension of New Shape/Dimension of Original Shape

Take an example of two squares having length-sides 6 unit and 3 unit respectively. Now, to find the scale factor follow the steps below.

Step 1: 6 x scale factor = 3

Step 2: Scale factor = 3/6 (Divide each side by 6).

Step 3: Scale factor = ½ =1:2(Simplified).

Hence, the scale factor from the larger Square to the smaller square is 1:2.

The scale factor can be used with various different shapes too.

Scale Factor Problem

For example, there’s a rectangle with measurements 6 cm and 3 cm.

Both sides of the rectangle will be doubled if we increase the scale factor for the original rectangle by 2. I.e By increasing the scale factor we mean to multiply the existing measurement of the rectangle by the given scale factor. Here, we have multiplied the original measurement of the rectangle by 2.

Originally, the rectangle’s length was 6 cm and Breadth was 3 cm.

After increasing its scale factor by 2, the length is 12 cm and Breadth is 6 cm.

Both sides will be triple if we increase the scale factor for the original rectangle by 3. I.e By increasing the scale factor we mean to multiply the existing measurement of the rectangle by the given scale factor. Here, we have multiplied the original measurement of the rectangle by 3.

After increasing its scale factor by 3, the length is 18 cm and Breadth is 9 cm.

How to find the scale factor of Enlargement

Problem 1: Increase the scale factor of the given Rectangle by 4.

Hint: Multiply the given measurements by 4.

Solution: Given Length of original rectangle = 4cm

Width or breadth of given rectangle = 2cm

Now as per the given question, we need to increase the size of the given triangle by scale factor of 4.

Thus, we need to multiply the dimensions of given rectangle by 4.

Therefore, the dimensions of new rectangle or enlarged rectangle is given by:

Length = 4 x 4 = 16cm

And Breadth = 2 x 4 = 8cm.

Scale Factor of 2

The scale factor of 2 means the new shape obtained after scaling the original shape is twice of the shape of the original shape.

The example below will help you to understand the concept of scale factor of 2.

Problem 2: Look at square Q. What scale factor has square P increased by?

Hint: Work Backwards, and divide the measurements of the new triangle by the original one to get the scale factor.

Solution : Divide the length of one side of the larger square by the scale factor.

We will get the length of the corresponding side of the smaller square.

The answer is 2.

The triangles which are similar have same shape and measure of three angles are also same. The only thing which varies is their sides. However, the ratio of the sides of one triangle is equivalent to the ratio of sides of another triangle, which is called here the scale factor.

If we have to find the enlarged triangle similar to the smaller triangle, we need to multiply the side-lengths of the smaller triangle by the scale factor.

Real-life Applications of Scale Factor

It is important to study real-life applications to understand the concept more clearly:

Because of various numbers getting multiplied or divided by a particular number to increase or decrease the given figure, scale factor can be compared to Ratios and Proportions .

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Scale Factor — Definition, Formula & How To Find

Scale factor definition

A scale factor in math is the ratio between corresponding measurements of an object and a representation of that object. If the scale factor is a whole number, the copy will be larger. If the scale factor is a fraction, the copy will be smaller.

A scale factor ratio can be expressed as a fraction, 1 2 \frac{1}{2} 2 1 , or a colon, 1:2 .

A ratio measures the relationship between two things. You could create a ratio of left-handed students to all students, but that ratio is not a scale factor.

How to find scale factor

To find the scale factor, you first decide which direction you are scaling:

Scale Up (smaller to larger) = larger measurement / smaller measurement

Scale Down (larger to smaller) = smaller measurement / larger measurement

The scale factor for scaling up is a ratio greater than 1 .

The scale factor for scaling down is a ratio less than 1 .

Once you know which way you are scaling, you compare corresponding sides using the correct basic equation. Compare the side length of the real object to the length of the corresponding side in the representation.

Finding scale factor of similar figures

Since we are scaling up , we divide the larger number by the smaller number:

The scale factor is 3 . To go from legs of 12 cm to legs of 36 cm , we needed to multiply 12 cm times 3 .

Now, let's try to scale down. Here are two similar pentagons. What is the scale factor used to create the second, smaller figure?

Scale factor of similar figures - pentagon scaling down example

Because we are scaling down, we divide corresponding side lengths (smaller number by larger number):

The scale factor is 1 7 \frac{1}{7} 7 1 . To get the second, smaller figure, we multiply 21 × 1 7 21\times \frac{1}{7} 21 × 7 1 ; the figure on the right uses a scale factor of 1:7 , 1 7 \frac{1}{7} 7 1 , or one−seventh .

Let's look at one more example and scale both up and down. Consider these two similar right triangles with labeled sides.

Finding scale factor of similar figures - right triangles example

If we have the little right triangle above and want to scale it up to the larger triangle, we write this:

The scale factor of the right triangle is 5:1. So every other linear measure is multiplied by 5 to scale them up.

If we have the big right triangle and want to scale it down to make the smaller one, we write this:

The scale factor of the right triangle is 1:5. So every other linear measure is multiplied by 1 5 \frac{1}{5} 5 1 to scale them up.

Scale factor in geometry

Scale is used in geometry to make accurate reproductions of figures; they are different sizes but not proportion. Figures are similar but to scale.

Scale factor is used on similar geometric figures. You can find the scale factor of corresponding angles, sides, and even diagonals.

How to reduce a shape by a scale factor?

Suppose you are given a figure and told to reduce it by 25% . Think in steps:

Are you making a larger or smaller dilation?

You are shrinking the original, so your scale factor will be less than a whole number.

Next, measure (or read) any side of the figure and do some math.

Suppose we have a rectangle that is 16 in. wide and we need to reduce it by 25% , or one-quarter ( 1 4 \frac{1}{4} 4 1 ).

That means it will be 75% of the original ( 100% − 25% = 75% ). We will use or 3:4 as our scale factor.

Multiply 16 × 3 4 \frac{3}{4} 4 3 :

Now, we simplify our answer:

The width of our smaller new shape must be 12 in. . We repeat these steps with the other dimension, 6 in. :

The height of our smaller rectangle must be 4.5 inches .

How to make a scale model

A scale model is a model accurate to a scale factor. If the copy of the actual object is not made to scale, it will look unrealistic, like a little child's toy.

One object can have different scales too. The greater the difference between the two numbers of the ratio, the smaller the model will be. A model that is 1:87 is generally going to be a lot smaller than a model with a ratio of 1:12 .

To make scale models, you need accurate plans of the original item, like a scale drawing . A scale drawing is an accurate plan of the real object, drawn using a scale factor to make the drawing small enough to handle.

You multiply every printed dimension on the scale drawing by your scale factor to get the right sizes for model parts. If, for instance, you wanted to build a simple shed for your model railroad scene, you would use the ratio 1 87 \frac{1}{87} 87 1 , so a 32-foot long shed would come out 4.4 inches long!

Scale factor examples

Try your hand at these questions to see if you understand the concept of scale factor in mathematics. Don’t shrink from it! Make an outsized effort!

What is the definition of scale factor?

How do you find the scale factor of similar figures?

What information does a scale factor give?

Define a scale drawing.

Please do not peek ahead until you try your best to find the answers.

The definition of scale factor is that it is a number that multiplies times a given quantity to produce a smaller or larger version of the original number. It is the ratio of a drawing, map, model, or blueprint to the actual object or distance.

You calculate the scale factor of similar figures by taking the ratio of corresponding parts of the two figures. When enlarging the shape, the larger measurement is the numerator, and the smaller measurement is the denominator. When shrinking the shape, the smaller measurement is the numerator, and the larger measurement is the denominator.

A scale factor gives the ratio of the representation to the actual object.

A scale drawing is an accurate drawing of an object done using a scale factor to shrink the original object's dimensions.

How to use scale factor

Scaling an object helps you visualize large real-world objects in small spaces or enlarge a small object for better viewing. Scale factor is how we ensure the representation of the object differs only in size from the original object.

We use scale to:

Draw similar figures in geometry

Make scale models

Draw scale blueprints of architecture and machinery

A common real-world use of scale factor is to bring vast areas of land down to small pieces of paper, like on a map.

Scale is used to allow designers, architects, and machinists to handle models of objects that would be too big to keep on a if they were actual size.

How Do You Solve a Scale Model Problem Using a Scale Factor?

This tutorial provides a great real world application of math! You'll see how to use the scale on a house blueprint to find the scale factor. Then, see how to use the scale factor and a measurement from the blueprint to find the measurement on the actual house! Check out this tutorial and see the usefulness blueprints and scale factor!

Background Tutorials

Fraction basics.

What's a Numerator and What's A Denominator?

Numerators and denominators are the key ingredients that make fractions, so if you want to work with fractions, you have to know what numerators and denominators are. Lucky for you, this tutorial will teach you some great tricks for remembering what numerators and denominators are all about.

How Do You Reduce A Fraction?

Fractions involving large numbers can be a handful, but sometimes these fractions can be reduced, taking those large numbers off your hands. This tutorial shows you how to reduce a fraction to its simplest form. Take a look!

Proportion Definitions

What's a Ratio?

Ratios are everywhere! The scale on a map or blueprint is a ratio. Ingredients sometimes need to be mixed using ratios such as the ratio of water to cement mix when making cement. Watch this tutorial to learn about ratios. Then think of some ratios you've encountered before!

Rates and Dimensional Analysis

How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate?

Word problems are a great way to see math in action! In this tutorial, learn how to use the information given in a word problem to create a rate. Then, find and use a conversion factor to convert a unit in the rate. Take a look!

Similar Figures and Indirect Measurement

What is a Scale Drawing?

Without a blueprint, it would be really hard to construct a building. Without a road map, you'd be lost! Scale drawings make it easy to see large things, like buildings and roads, on paper. Even a GPS uses scale drawings! Check out this tutorial to learn all about scale drawings.

How Do You Find Missing Measurements of Similar Figures Using a Scale Factor?

How Do You Find a Scale Factor in Similar Figures?

Further Exploration

Working with proportions.

How Do You Solve a Word Problem Using a Proportion?

This tutorial provides a great real world application of math. You'll see how to use the scale from a blueprint of a house to help find the actual height of the house. This tutorial shows you how to use a proportion to solve!

Scale Factor of Similar Figures Worksheets

Have you ever wondered, how huge objects are created from miniature blueprints??? This collection of printable worksheets assists the 7th grade, 8th grade, and high school students in comprehending scale factor of similar figures. Few instances of real-life application of scale factor are creating miniature models, blueprints and engineering designs. An array of skills like drawing new shapes, enlarge (up scale) or reduce (down scale) shapes, find the missing sides, word problems and more are encompassed here. Begin your practice with our free scale factor worksheets!

Enlarge or Reduce - Shapes

Get a better understanding of the concept by enlarging or reducing the shapes using the given scale factors.

Enlarge or Reduce - Real Life Objects

This set of 7th grade scale factor worksheet pdfs features fascinating real-life pictures like house, rocket, Christmas tree and more. Up scale or down scale the image according to the scale factor and draw the new image.

Scale Factor - MCQ

Put on your thinking caps to find the answer that best fits the problem in these printable MCQ worksheets. Determine the actual length, find the original or scaled copy of a model, identify the scale factor of similar figures and more.

Scale Factor Word Problems - Level 1

The scale factor word problems here encompass attractive pictures with real-life scenarios to determine the parameters like length, width, distance for the model or real objects.

Scale Factor Word Problems - Level 2

Application of scale factor in the real-world context is structured into level 2 word problems. Learners in grade 7 and grade 8 are required to find the scale factor of the real or dilated image and their corresponding linear measurements.

Find the Missing Side - Level 1

The Level 1 worksheets consist of similar shapes with scale factors in whole numbers. Determine the value of the labeled sides using the given scale factor.

Find the Missing Side - Level 2

Mixed Review - MCQ

Recapitulate the knowledge acquired by implementing these all-inclusive, mixed review MCQ worksheets for high school to find the scale factor of similar figures, determine the ratio of areas, perimeters, surface areas and volumes, dilation and more.

Scale Factor Worksheets - Area and Perimeter

This set of printable worksheets assists students to learn how the scale factor of similar figures influences the ratio of areas and perimeters.

(30 Worksheets)

Scale Factor Worksheets - Surface Area and Volume

Employ this ideal set of worksheets that consist of an array of skills like finding the scale factor, the ratio of surface areas, the ratio of volumes, word problems related to solid shapes and much more.

(33 Worksheets)

Dilation Worksheets - Center at the Origin

Offering a blend of exercises, these dilation - center at the origin worksheets, contain tasks like identifying the type of dilation, writing the scale factor, finding the dilated coordinates and using them to draw the dilated images.

(18 Worksheets)

Dilation Worksheets - Center not at the Origin

Introduce the concept of dilation with exercises like writing the coordinate rule, finding the dilated coordinates and drawing dilated shapes with this extensive collection of worksheet pdfs with the center not at origin.

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COMMENTS

4 Ways to Find Scale Factor
Use the scale factor to answer this problem. Multiply the height of Rectangle ABCD by the scale factor. This will give you the height of Rectangle EFGH: 3 × 2.5 = 7. Multiply the width of Rectangle ABCD by the scale factor. This will give you the width of Rectangle EFGH: 8 × 2.5 = 20... Multiply the ...
Scale Factor
How to find the scale factor of Enlargement Problem 1: Increase the scale factor of the given Rectangle by 4. Hint: Multiply the given measurements by 4. Solution: Given Length of original rectangle = 4cm Width or breadth of given rectangle = 2cm Now as per the given question, we need to increase the size of the given triangle by scale factor of 4.
Scale Factor
To find the scale factor, you first decide which direction you are scaling: Scale Up (smaller to larger) = larger measurement / smaller measurement. Scale Down (larger to smaller) = smaller measurement / larger measurement.
How Do You Solve a Scale Model Problem Using a Scale Factor?
You could use a scale factor to solve! In this tutorial, learn how to create a ratio of corresponding sides with known length and use the ratio to find the scale factor. Then, write an equation using the scale factor to find your missing measurement! How Do You Find a Scale Factor in Similar Figures? Have similar figures?
Solving a scale drawing word problem (video)
Solving a scale drawing word problem CCSS.Math: 7.G.A.1 Google Classroom About Transcript See how we solve a word problem by using a scale drawing and finding the scale factor. Created by Sal Khan. Sort by: Top Voted Questions Tips & Thanks shillpak28 7 years ago at 3:47
Scale Factor Worksheets
Scale Factor - MCQ Put on your thinking caps to find the answer that best fits the problem in these printable MCQ worksheets. Determine the actual length, find the original or scaled copy of a model, identify the scale factor of similar figures and more. Scale Factor Word Problems - Level 1
How to Solve Word Problems Involving Scale Drawings
Step 1: Find a common length whose measurements are given both on the drawing or model, and in real life. Step 2: If necessary, set these two measurements equal to each other using on-the-drawing (OTD), and in-real-life (IRL)... Step 3: Divide both sides of the equality by the number of ...
Solving For Scale Factor
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Scale factor problems
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Solving One Variable Equations Scale Model Problems Mr. T's Math Videos 6.53K subscribers Subscribe 163 33K views 13 years ago 2 problems showing how solve …