# Loopy Polygons

**'Fit polygons together to make loops. Discover the patterns.'**

Octagons can be fitted together to make loops. Try it for yourself with this applet!

How many octagons do you need to make a loop? Are there certain numbers of octagons that will **not** make a loop? What is the inside perimeter of the shape? What is the outside perimeter? The perimeters form and interesting pattern. Can you find a formula that links the perimeter and the number of octagons?

Can you make loops with other polygons? What are the properties of the polygon that mean that it can form loops?

All these questions and many more can be explored in this investigation about **loopy polygons**.

*This investigation is based on an old GCSE coursework, 'Octagon Loops'.*

The following animation should give an idea of what the investigation entails:

### The Rules

**Here are the only important rules for making loops from polygons**

**1. ****Edges must match up exactly**

**2. ****Only one space can be created in the middle of the loop**

**3. ****Each polygon must touch exactly two others**

### Instructions

This investigation can be attempted with just the animation and the rules above.

A more structured investigation can be completed using the following resources:

- Here is a structured investigation where students are encouraged to complete tables and draw graphs to find the formula.
- Tiles that can be cut out and manipulated by printing the following: octagons hexagons squares triangles
- Students can be given help with writing up their investigation.

Click on the following photos to see some students working on the investigation in class:

### Manipulatives

Here are some applets that can be used to explore the loops in this investigation. Paper versions of these can be printed out from above.

#### Octagon Loops

You may wish to use this manipulative to make loops with equilateral octagons.

The bottom right vertex rotates the octagon.

#### Hexagon Loops

You may wish to use this manipulative to make loops with equilateral hexagons.

The bottom right vertex rotates the hexagon.

#### Square Loops

You may wish to use this manipulative to make loops with squares.

#### Triangle Loops

You may wish to use this manipulative to make loops with equilateral triangles.

The bottom right vertex rotates the triangle.

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