Scale Drawings

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:

10\div\dfrac{1}{15}=150\text{ cm}

 

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}