# Triangles all Around

You might like to have a look at Nine-Pin Triangles before trying this problem.

How many different triangles can you draw on a circular pegboard which has four equally spaced pegs?

What are the angles of each triangle?

If you have a six-peg circular pegboard, how many different triangles are possible now?

What are their angles?

How many different triangles could you draw on an eight-peg board?

Can you find the angles of each?

You may like to use the interactivity to try out your ideas. When you have selected the number of dots you need, select the line drawing tool, then click on two dots to draw a line between them.

For printable sets of circle templates for use with this activity, please see our Printable Resources page.

*Many thanks to Geoff Faux who introduced us to the merits of the nine-pin circular geoboard.*

*For further ideas about using geoboards in the classroom, please see Geoff's publications available through the Association of Teachers of Mathematics (search for 'geoboards').*

How are you deciding what is the same and what is different?

What do you know about the angles in a triangle?

It might help to mark the centre of the circle and to draw in some radii.

To solve this problem I made all the dots connect with lines creating the maximum amount on triangles possible in each different circular pegboard.

On the 4 pegboard, I made 4 right angle triangles.

On the 6 pegboard, I made 18 triangles but I couldn't find out which triangles were there.

On the 8 pegboard, I made about 56 triangles, I may have counted wrong, but I can promise there were lots of triangles.

Georgia, also from Woodlea Junior School, wrote:

I knew that if you halve a diamond (or a square) it makes a triangle, so the first thing that I did was make a diamond on the circle. Then I made another diamond that was facing the other way round. By then, I already had about 15 triangles. After that, I turned on the centre point and started making lots of little triangles by connecting two lines in the centre so that they met in the middle and made a triangle shape. I did that over and over again, but each time I made sure that I wasn't ruining any of the other triangles unless when I did, it just split it into two smaller triangles. When I'd finished doing that, I couldn't do anymore because if I did, it would have made lots of the triangles turn into shapes that weren't triangles.

Nico from Toucan in Panama wrote:

I started by drawing squares inside the circle, then divided them by four. Each square gives you four triangle. Each peg should have 4 straight lines coming out from them.

Roberto also from Toucan in Panama said:

If you draw 4 straight lines out of each peg you make 4 triangles. In the end, you end up with 32 triangles out of each peg.

Thank you all. I wonder whether you thought about what makes one triangle different from another one... For example, on the four-peg board, Chloe says she created four triangles. Are they all different from each other?

Sam sent us her work on the problem, including the angles of all the triangles. Thank you Sam!

Can you see how she avoided counting any twice?

She first counted triangles with two corners on neighbouring pegs, then those two apart, and so on. She called two triangles the same if they had the same angles and not just if they used the same pegs. She explains this in a bit more detail below.

Here is her work:

First I labelled the points on the pegboard $ABCD$.

There are four possible triangles: $ABC$, $ABD$, $ACD$ and $BCD$. However these triangles are all the same shape (you can see this by rotating triangle $ABC$) so we could say that there is only one type of triangle that we can make.

This triangle has angles $90^\circ$, $45^\circ$ and $45^\circ$. I know this because if you draw a square around the points $ABCD$ and cut it in along the diagonal you get this triangle.

For the six point board, I again labelled the points as $ABCDEF$.

There are three possible triangles

$ABC$, with angles $120^\circ$, $30^\circ$ and $30^\circ$.

$ABD$, with angles $90^\circ$, $60^\circ$ and $30^\circ$.

$ACE$, with all angles $60^\circ$ (an equilateral triangle).

For the eight point board, I again labelled the pegs $ABCDEFGH$

There are five possible triangles

ABC, with angles $135^\circ$, $22.5^\circ$ and $22.5^\circ$.

ABD, with angles $112.5^\circ$, $22.5^\circ$ and $45^\circ$.

ABE, with angles $90^\circ$, $22.5^\circ$ and $67.5^\circ$.

ACE, with angles $90^\circ$, $45^\circ$ and $45^\circ$.

ACF, with angles $45^\circ$, $67.5^\circ$ and $67.5^\circ$.

Can you see how she worked out the angles in the triangles? Remember that the angles in a triangle add up to 180 degrees! You can divide the triangle (or the circle) into pieces whose angles you know to help you.

Why do this problem?

This problem is designed to provoke learners' curiosity. Some may be intrigued by what constitutes a 'different' triangle; others might be keen to know they have found all the possibilities for a given pegboard. The interactivity will help learners try out their ideas and deepen their understanding of the concept of a triangle. The task offers an opportunity for pupils to work in a
systematic way, using their knowledge of the properties of triangles, angles in circles and angles in triangles. (A knowledge of circle theorems is not required.) The problem encourages learners to be clear about what they do know and what they can work out from it.

#### Possible approach

It would be a good idea to try Nine-pin Triangles before tackling this task. You may like to read the teachers' notes of that task and follow a similar approach.

After learners have been working on the four-peg board for a short while, it would be helpful to facilitate a discussion about which triangles are the same and which are different, and why. Once the possibilities have been shared, it might be useful to suggest that if two triangles could be picked up and placed exactly on top of each other, then they are the same (i.e. that orientation on
the pegboard does not matter). This makes the total number to be found more manageable in each case.

If working on paper rather than using the interactivity, pupils may find it helpful to print these sheets off:

Sheet of four-peg boards

Sheet of six-peg boards

Sheet of eight-peg boards

In addition to being a useful tool for working on the problem, the interactivity will allow you to share learners' findings during a final plenary. You could warn a few pairs that you'd like them to draw their triangle/s on the board using the interactivity and to explain how they calculated the angles. It would be interesting to see whether other pairs had different methods for working out
the angles. Another pair could be chosen to justify the total number of triangles they had found for a particular size pegboard.

#### Key questions

How do you know your triangles are all different?

How do you know you have got all the different triangles?

What do you know about the angles in a triangle?

If you mark the centre of your circle and draw in some radii, how might that help?

How many degrees are there in a full turn?

#### Possible support

Working in pairs will help learners access this task. After some time, you could encourage two pairs to join forces and compare their ways of working.