Tri-Five

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

Age

7 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Find all the DIFFERENT shapes that can be made by joining five equilateral triangles edge to edge.

Here's an example:

Finding the best solution for this problem depended on thinking very carefully about what was meant by DIFFERENT ways of arranging the five triangles. For example: Christine (Malborough School) explained:

There is only one shape from the hexagon group because when this shape is rotated it looks like different shapes but it is just one basic shape.

As Sophie said:

There were four different shapes using all five equilateral triangles, without rotating the shapes. The shapes I made were:

One was long and thin, but was not a perfect line because it had three triangles on one side and two triangles on the top side.

The second shape had four triangles sloping upwards, and one triangle on the side of it.

The third shape was a big triangle with a smaller triangle on the side of the top one.

And the fourth shape was shaped in the way of a croissant, and had three triangles in a row, and two on top of the two end triangles.

Leyla (Private IRMAK Primary School, Istanbul, Turkey) sent in drawings of the four ways:

Merve (Private IRMAK Primary School, Istanbul, Turkey) agreed with this set of four shapes too.

Caroline and Rebecca (The Mount School, York) also realised that some shapes they found were really the same as others if you turned them around or flipped them over:

We found six different ones, but two of these are reflections in a way, so maybe it's only four.

Kirstine (Tattingstone School) also saw how to group some 'variations' of the same shape together. However, if you decided to think about each position of the shape as being different, then there would be many shapes in your solution.

Ece (Private Irmak Primary School) found $18$ shapes. Do think there are any more?

Christopher (Tattingstone School) found two more variations of the straight line. Well done to everyone else who sent in some shapes.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners

Tri-Five

Tri-Five

Why do this problem?

Possible approach

Key questions

Possible extension

Possible support