# Scale Drawings

## Scale Drawings Revision

**Scale Drawings**

A **scale drawing** portrays a real object with the lengths in proportion to the lengths of the actual object, but reduced or enlarged by a **scale factor**.

You may be asked to find the **scale (or scale factor)**, and/or **calculate actual distances** based on a **scale drawing**.

**Scales and Scale Factors**

Below is a scale drawing of a tennis court with a **scale** 1\text{ cm}:4\text{ m} or 1:400.

We can use a **scale** of 1:400 to calculate the **scale factor**, by dividing the left hand side of the ratio by the right hand side:

**Scale factor**: 1\div400=\dfrac{1}{400}

This means, the actual tennis court size has been enlarged by a **scale factor** of \dfrac{1}{400} to give the scale drawing.

If we did not have the **scale**, we could work out both the **scale** and the** scale factor** from the drawing:

**Scale**: we can use any length from the scale drawing and actual object to calculate this, making sure to choose the equivalent side in both.

Length scale drawing : Length actual object

6\text{ cm}:24\text{ m}\\ 6\text{ cm}:2400\text{ cm}\\ 1\text{ cm}: 400\text{ cm}\\ 1:400

**Scale factor**: again, choose equivalent lengths from the scale drawing and actual image.

Length scale drawing \div Length actual drawing

6\div2400\\ 1\div400\\ \dfrac{1}{400}

**Calculating Lengths**

We can use a **scale** or **scale factor** to **calculate lengths** or **distances** when given only a scale drawing.

Here is a scale drawing of a rug with **scale** 1:15, calculate the **length** and **width** of the actual rug.

**Length**: the **length** of the scale drawing is 10\text{ cm} and the **scale** is 1:15.

The 1 in this ratio represents the scale drawing, and the 15 represents the actual rug, so the actual rug is 15 times the size of the scale drawing.

Therefore, the **length** is 10\times15=150\text{ cm}

**Width**: we can do the same here but using the value of the scale drawing’s width: 5\text{ cm}.

Therefore, the width is 5\times15=75\text{ cm}

**Note**: if we were told instead that the **scale factor** of the actual rug to the image is \dfrac{1}{15} here, to find the **length** we would:

**Drawing Scale Diagrams**

You may be asked to draw a scale drawing based off actual dimensions.

For example:

Below is an image of a rectangular door showing its actual dimensions. Given that the **scale** is 1:8, draw a scale drawing of the door.

To draw a scale drawing, you firstly need to work out the length of the sides of the drawing using the method we learnt previously:

As we have the actual object’s size, we will divide the width and height by 8 to find the scaled lengths:

Width: 80\div8=10\text{ cm}

Height: 200\div8=25\text{ cm}

Therefore, we need to draw a rectangle with width 10\text{ cm}, and height 25\text{ cm}:

Where the squares in the paper are 1\text{ cm}

**Example 1: Scales and Scale Factors**

Below is a scale drawing of a box in comparison to the actual box. Calculate the **scale** of this drawing.

**[2 marks]**

To calculate the **scale**, we need to find a ratio:

Length scale drawing : Length actual object

2\text{ cm}:12\text{ cm}\\ 1:6**Note**: we could have used the heights of this boxes to calculate this and it would have given the same answer.

**Example 2: Calculating Lengths**

Below is a scale drawing of a ball. The actual ball has been enlarged by **scale factor** \dfrac{1}{9}. Calculate the **diameter** of the actual ball.

**[2 marks]**

To calculate the diameter, we need to divide the scale drawing’s **diameter** by the scale factor:

## Scale Drawings Example Questions

**Question 1:** The following scale drawing represents an ice cream cone scaled by 1:5. Calculate the height of the actual ice cream cone.

**[2 marks]**

The scaled height is 3\text{ cm} and the scale is 1:5, so we multiply 3 by 5 to find the actual height:

3\times5=15\text{ cm}**Question 2:** A scale drawing of a bus has length 12\text{ cm}, while the actual bus’ length is 12\text{ m}, find the scale in its simplest form.

**[2 marks]**

The ratio of scaled to actual is:

12\text{ cm}:12\text{ m}Convert meters to centimeters:

12\text{ cm}:1200\text{ cm}We can then divide this by 12 to find the simplest form:

1:100**Question 3:** A map has scale 1:120000. The distance between Jemima’s house and school on the map is 8\text{ cm}. Find the actual distance between Jemima’s house and school, giving your answer in \text{km}.

**[3 marks]**

Firstly, multiply the scaled distance by the ratio:

8\times120000=960000\text{ cm}To convert \text{cm} to \text{km}, we divide by 100000:

960000\div100000=9.6\text{km}