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How to Make a Mathematical Paper Snowflake (Christmas Crafts)

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I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.

Mathematical symmetrical snowflake

Mathematical symmetrical snowflake

Mathematical Snowflakes

In this quick tutorial, we are going to look at how to make a very simple but effective symmetrical snowflake using only paper, a pencil, a pair of compasses, scissors and a straight edge/ruler.

These are great for homemade Christmas decorations and can be completed in all kinds of sizes and in various patterns. I've also used them as a Christmas-themed craft in maths lessons as the making of them uses several mathematical skills and is great for practising terminology such as reflective and rotational symmetry.

Ruler, pencil, paper, scissors and compasses

Ruler, pencil, paper, scissors and compasses

Materials Needed

The only materials you will need to make the snowflake are as follows:

  • a sheet of paper (any size will do)
  • a sharp pencil
  • a straight edge (I used a ruler, but you won't need to measure anything, so any straight edge will do)
  • a pair of scissors
  • a compass

Step 1. Drawing a Circle

The first step is to draw a circle using your pair of compasses. It doesn't matter how big you make your circle; the bigger the circle, the bigger the snowflake. The diameter of the completed snowflake will be the same as your circle.

Make sure that you don't close your compass after drawing the circle. The next step will require your compass to be at the exact same setting in order for the project to work.

Drawing a circle using a pair of compasses

Drawing a circle using a pair of compasses

Step 2. Make Equidistant Marks Around Edge of Circle

You should now have a completed circle on your page. Make a mark at one point on the circumference of this circle (it doesn't matter where) as in picture 1 below. Place the spike of your compass on this mark and draw an arc cutting through the circle further round the circumference as in picture 2 below.

You now need to move the compass to the point where this arc meets the circumference and repeat the last step, drawing a new arc even further around the circle. Keep on repeating this as in picture 3 below. If you have done all of this correctly, you should be able to place the compass on your final arc and have the pencil reach back to the original starting point. If this doesn't happen, check that your compass has remained at the same size by comparing them to the circle's radius.

You now need to take your pencil and straight edge, and draw straight lines linking each arc-circumference crossover to the next one as in picture 4 below. Once this is done you will end up with a regular hexagon, which now requires cutting out.

Making Equidistant Marks Around the Circle

Making Equidistant Marks Around the Circle

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Step 3. Cut Out Regular Hexagon

Regular Hexagon Created Without Measuring Lines or Angles

Regular Hexagon Created Without Measuring Lines or Angles

Step 4. Fold the Hexagon

We now need to do some folding in order to convert our hexagon into a snowflake with reflectional and rotational symmetry.

Fold your hexagon in half along a diagonal. You will end up with an isosceles trapezium as in picture 1 below.

Take the bottom left-hand corner and fold this up to the top right-hand corner so that the left-hand edge matches up with the top edge as in pictures 2 and 3 below. You should now have a rhombus/diamond.

Repeat this step, this time folding the bottom right-hand corner up to the top left so that you end up with an equilateral triangle as in picture 4 below.

Step 5. Fold the Hexagon

Folding the Hexagon Before Cutting the Snowflake

Folding the Hexagon Before Cutting the Snowflake

Step 6. Create a Snowflake Pattern

Now you have an equilateral triangle, the next step is to cut the snowflake pattern. Go around each of the three edges and carefully cut out whatever shapes you like. In the picture below I have cut out a variety of different sized triangles, but you can experiment with other shapes such as quadrilaterals.

Make sure that you don't completely remove any of the edges or your snowflake will fall apart at the next stage. You can see in the picture below that I have left gaps between each of the triangles.

Step 7. Cut a Pattern Into Your Triangle

Cutting a Pattern Into Your Triangle

Cutting a Pattern Into Your Triangle

Step 8. Complete the Snowflake

Once you have cut your shapes from the edges, simply open the triangle back out into a hexagon and you will have a finished snowflake. If at this point you feel like you need more detail in your snowflake, you can fold it back into a triangle again and do some more cutting.

The Finished Snowflake

Mathematical Christmas Snowflake

Mathematical Christmas Snowflake

What To Do Next

Once you have completed your first paper snowflake, why not try experimenting with more designs? You can change the size of the initial circle to create different-sized snowflakes or try cutting out different shapes at the triangle stage in order to create different patterns within your snowflake. You can even use coloured paper to create some variety in your snowflakes; pale blue works particularly well.

Making Mathematical Snowflakes in a Math Lesson

This is a great activity to give to a math class during the run-up to Christmas for a number of reasons.

  • It practises construction skills by requiring the accurate use of compasses, pencil and ruler.
  • You can get your class thinking about why the construction method gives a regular hexagon. Hint: It is to do with creating equilateral triangles, which is why it is so important to keep the compasses' setting the same all the way through.
  • You can get your class working out how many lines of reflectional symmetry or what order of rotational symmetry their snowflake has. Challenge them to make snowflakes with different orders of rotational symmetry.
  • It's also great for decorating the classroom! I always stick mine to the windows with blu-tac.

© 2022 David