Master

Scale Factor

Master scale factor with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Scale Factor

Study Guide

Key Definition

The scale factor is the ratio of one size to another in similar figures. For example, if two triangles are similar and one side of the larger triangle is $4$ times the corresponding side of the smaller triangle, the scale factor is $4$.

Important Notes

Scale factor is used to enlarge or reduce figures proportionally.
When dilating an object in 2D, the area increases by the scale factor squared: $k^2$.
For 3D objects, the volume increases by the scale factor cubed: $k^3$.
In similar figures, corresponding angles are equal and corresponding sides are proportional.
The scale factor can be expressed as a fraction, a decimal, or a whole number.

Mathematical Notation

$\angle$ denotes an angle

$\triangle$ denotes a triangle

$\cong$ means congruent

$\sim$ means similar

$\frac{a}{b}$ represents a fraction

Remember to use proper notation when solving problems

Why It Works

Scale factor works because it maintains the proportionality of figures, ensuring that all corresponding angles remain equal and all corresponding sides maintain the same ratio: $\frac{XY}{UV} = \frac{YZ}{VW} = \frac{ZX}{WU}$.

Remember

The scale factor helps in creating model representations of objects where dimensions are reduced or enlarged proportionally.

Quick Reference

Scale Factor Formula:$\text{Scale Factor} = \frac{\text{Dimension of Image}}{\text{Dimension of Original}}$

Understanding Scale Factor

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The scale factor is a simple ratio that describes how much a figure is enlarged or reduced. If a model is $\frac{1}{72}$ the size of the original, the scale factor is $\frac{1}{72}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the scale factor if a triangle with side length $3$ is enlarged to have a side length of $12$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are building a model car that is $\frac{1}{10}$ the size of a real car. If the real car is $4.5$ meters long, how long is your model?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a painting is enlarged by a scale factor of $3$, by what factor does the area increase?

Challenge Quiz

Single Choice Quiz

Advanced

A cube is enlarged by a scale factor of $2$. What is the factor of increase for its volume?

Recap

Watch & Learn

Review key concepts and takeaways