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# Scale|Definition & Meaning

## Definition

In math, the term **scale** defines the **ratio** of the drawn and the **actual** **units** of something. For example, if an inch in any direction on a map **represents** a quarter of a mile, we say that the map is drawn to a **scale** of 4 inches to a mile, or 4 inches: 1 mile. **Scales** are especially useful with **graphs** since we can represent **functions** with large values or jumps in a small region.

## What Is a Scale?

Scale is the **factor** that gives the ratio between the length of the actual figure and drawing its portrait. Scale gives the measurement of how much the portrait or **model** is in resemblance to the real thing. Scale is much used in **sketching** and drawing. **It may be expressed in ratios, words, and fractions.**

Ever wondered how maps of huge-sized buildings and sculptures are constructed with precise **measurements** and everything accurately designed? Scaling helps in the cause of mapping such huge **measurements** to a small piece of **paper,** so it becomes easy to understand and convey to others.

In mathematics, scaling is a method in which an actual **size **object is narrowed down or scaled up in exact proportions so that it can be to be drawn on a piece of **paper** or whiteboard. In geometry, **scaling** refers to more or less **extending** or **shortening** structures in order to preserve their fundamental **form**. Identical figures are those which have been scaled.

In the above figure, both shapes are similar to each other, but one rectangle is **downsized,** whereas the other one is in its actual shape. So to narrow down or shrink any **large** scaled **sculpture**, we take the use of a **scale**.

## How To Scale Using a Scale Factor?

A **scale** is generally used to represent two **models** or **images** using a common **relationship** or by representing them in a ratio of their interconnected dimensions. Usually, the dimensions of the actual **image** are factored into another measuring **scale**. This measuring **scale** contains a common number known as the **scale** **factor.**

The **scale factor** is responsible for enlarging or shrinking an image. It is multiplied by all the components of the actual **model** resulting in a scaled image. A measuring **scale** and the **scale** **factor** are most commonly used in blueprints that are helpful in mapping construction sites.

Let’s suppose we need to construct an **image** of an eighty feet tall tree on a piece of **paper**. For this miracle to happen, we need to define an appropriate **scale** that fits our **paper** and accurately define our **measurements**.

We say that 10 feet are equal to a 1-inch **scale** which can be represented as 1:10, and in **mathematical** **terms**, it means that 1 unit in drawing perspectives will represent 10 units in the actual perspective.

So a tree that tall can easily fit in our **paper** when we **scale** it down to only 8 inches. So in the real world, the tree is 80 feet tall, but it can be represented as 8 inches on a piece of **paper**, thus indicating that a **scale** of 1:10 has been applied here.

If there is not much information about the **scale** **factor** in a given set of **images**, then by knowing the actual dimensions of the original **model** and the drawn image, we can easily calculate the **scale** **factor**.

The **formula** to find the **scale** and the **scale** **factor** is as follows:

**Measuring Scale = Measurements of scaled image : Measurements of the actual image**

**Scale factor = Measurements of scaled image / Measurements of the actual image**

To better understand the **formulas**, let’s say a lamp pole is to be made that is 100 inches high in the **absolute** **world** but is portrayed as only 10 inches in the **drawing**. So the **scale** can be determined as 10:100 or 1:10, and the **scale** **factor** will be 1/10.

## Scaling Types

### Up-Scaling

**Up-scaling** means to enlarge something. When we want to view something in enlarged form, we **scale** up the smaller figure. In simple words, when the **size **of an image is increased as compared to its actual **size**, it indicates that the image has been scaled up. The **formulas** for **scale** and scale-up **factor** are constructed as:

**Measuring Scale = Measurements of scaled image : Measurements of the actual image**

**Up-Scale factor = Enlarged image dimensions / Dimensions of the small image**

In situations where the image is really compact, we may want to **scale** it up. To analyze a mosquito’s wings, for instance, we must sketch them out on canvas.

We would want to ramp up the **mosquito** because of its tiny **dimension**, as illustrated here.

Here, the **scale** **factor** is:

$\dfrac{15}{3}$ = 5 > 1

As you can see, the **scale** **factor** for **enlarging** an image will always be bigger than 1. Let’s say we have another scenario where one image is 4 times larger than the smaller **image**, so we get the **scale** **factor** as 4 since the ratio is 1:4, thus making it **greater** than 1. The smaller **models** are referred to as **preimages,** whereas the enlarged **model** is known as an image.

### Down-Scaling

Down-scaling means reducing the **size **of something. When we want to view a big figure in a smaller form, we **scale** down the large figure. In simple words, when the **size **of an image is decreased as compared to its actual size, it indicates that the image has been scaled down. The **formulas** for **scale** and **scale** down **factor** are formed as:

**Scale = Measurements of scaled image : Measurements of the actual image**

**Down-Scale factor = Reduced image dimensions / Dimensions of the Larger Image**

In situations where the object is really big, we may want to **scale** it down. To construct a master bedroom, for instance, we must sketch it out on **paper**.

Let’s say the actual **size **of the bedroom is 240 inches by 210 inches. Since it’s rather a difficult task to draw it on **paper**, we must **scale** it down to a drawable **size, **such as 8 inches by 7 inches.

Here, the **scale** **factor** is:

$\dfrac{8}{240}$ = $\dfrac{1}{30}$ < 1

As you can see, the **scale** **factor** for shrinking an image will always be smaller than 1.

Let’s say we have another scenario where one image is 4 times **smaller** than the larger image, so we get the **scale** **factor** as 1/4 since the ratio is 4:1, thus making it smaller than 1. The smaller **models** are referred to as **images,** whereas the actual **model** is known as a **preimage**.

## Solved Examples Related To Scale

### Example 1

Determine the length of an actual **sculpture** if the length of the scaled figure is 5 units and the **scale** ratio is 1:25.

### Solution

\[ \dfrac{\text{Scaled Height}}{\text{Actual Height}} = \dfrac{1}{25}\]

\[ \dfrac{5}{\text{Actual Height}} = \dfrac{1}{25}\]

Actual Height = 5 $\times$ 25 = 125 length units

### Example 2

A triangle now has new **dimensions** of 6 cm by 10 cm by 12 cm after being scaled up by a **scale** **factor** of 2. What will be the **original** **dimensions** of the triangle?

### Solution

Given new dimensions, 6 cm, 10 cm, and 12 cm, and the **scale** **factor** of 2, the **formula** for the **scale** **factor** is:

Scale factor = $ \dfrac{\text{Measurements of scaled image}}{\text{Measurements of the actual image}} $

Substituting the values for all three sides gives:

- 2 = $\dfrac{6}{\text{Actual Dimensions}}$

Actual Dimensions = $\dfrac{6}{2}$ = 3 cm

- 2 = $\dfrac{10}{\text{Actual Dimensions}}$

Actual Dimensions = $\dfrac{10}{2}$ = 5 cm

- 2 = $\dfrac{12}{\text{Actual Dimensions}}$

Actual Dimensions = $\dfrac{12}{2}$ = 6 cm

Hence the **original** dimensions of the **triangle** are 3 cm, 5cm, and 6 cm.

*All images are created using GeoGebra. *