# 10.3 Scale Diagrams

This section considers how you can use **scale diagrams** in practical
situations. Scale diagrams are particularly useful when you want to describe
something accurately, but the life-size version is either too big or too small
to do so directly. For example, a map can highlight key features of the land and
help organize a trip, or a plan can indicate how to build something to
specifications. In a scale diagram, all the measurements are a fraction (or a
multiple) of the real measurements. For example, suppose you have a plan for a
table, and a scale of 1:50 is used. This means that the real measurements will
be 50 times the measurements on the plan. So if a particular length on the plan
is 4 cm, the corresponding length on the full-size table would be
.

To understand how this works in practice, try making a geometrical puzzle
from a scale diagram in the activity below. This kind of puzzle is known as a
**tangram** and has been popular in many countries for hundreds of years.
It consists of seven geometrical pieces arranged in a square, and the challenge
is to arrange all seven pieces to make another shape or picture that is given to
you. You will need the tangram for activities in sections 10.4 and 10.5. [
The earliest known book of tangram puzzles was published
in China in 1813.
]

You will need to print out the tangram for the following activity. Use the link provided below.

The following activity requires you to print out a tangram. Use the link below to download and print a copy of the tangram:

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## Activity: Making a Tangram Puzzle

A diagram of the tangram puzzle is shown below. Use the similar tangram that you have just printed out.

Note that the image you see on-screen may not appear drawn to the proper scale.

(a) Draw the square outline of the full-size puzzle on a thin cardboard, at a ratio of 1:2. Use your ruler to enlarge the outline of the printed tangram.

Measure the length of the side of the entire puzzle you printed out. What do you need to use for the corresponding lengths on the resized puzzle?

### Comment

You may find it helpful to place a piece of graph paper onto the cardboard first, so that your lines are straight and the corners square.

### Answer

(a) If you scaled your tangram properly, the length of each side of the tangram will double to 4 inches.

(b) Add the grid of 16 squares to the puzzle by drawing the three evenly spaced horizontal vertical lines. What are the side lengths of the 16 small squares in your scaled diagram?

### Answer

(b) The lengths of the small squares on the new puzzle should be twice the length on the original diagram.

The length of the sides of the small squares in the original puzzle are 0.5 in. The corresponding lengths on the new puzzle are therefore .

Finally, add the thick blue construction lines by matching the points of intersection. Keep your puzzle handy, but don’t cut it up yet—there’s more to be done!

You may have used scale diagrams for do-it-yourself work in your house or garden. If you are planning a new look for a room or a garden, it helps to create a detailed plan so that you can see how things will look. Bathroom suppliers and do-it-yourself stores often provide planning grids and scaled cut-out shapes for items such as bathtubs and cupboards, so you can try different arrangements without moving the real furniture. A scale diagram also helps you to see what will fit and more importantly what won’t, and this can prevent expensive mistakes.

Imagine a planning grid marked in 0.5 cm squares with scale at 1:20. What would each square on the plan represent in real life?

Since the scale is 1:20, the side of each of these squares will represent , or 10 cm.

In order to use such a grid, you will need to add the dimensions of the room and the measurements of fixtures like doors, windows, and radiators. For example, suppose you measure your bathroom and find that it is 3.5 meters long and 2.45 meters wide. So that the diagram will fit onto a standard letter-size piece of paper, use either centimeters or millimeters as the units of measurement.

- First, convert the real-life measurements into centimeters. The length of the bathroom is .
- The edge of each square on the plan represents 10 cm.
- Since , the edge of 35 squares will represent 350 cm.
- The width of the bathroom is 2.45 m .
- Since , we know that the edge of 24.5 squares will represent 245 cm.

On the planning grid provided separately, draw a rectangle 35 squares long and 24.5 squares wide to mark the boundary of the bathroom.

Use the link below to download the bathroom planning grid:

Note: when printing, be sure to place your settings on “Scale: None” to ensure that the dimensions print accurately.

Alternatively, rather than counting squares, you can calculate the lengths on the plan. If the scale is 1:20, then you will have to divide both the length and the width of the bathroom by 20 to find the corresponding lengths on the plan. The length of the bathroom will be , and the width of the bathroom will be . Check that your rectangle measures 17.5 cm by 12.25 cm!

10.2.3 Calculator Exploration: Cube Roots and More