Triangle transformation
Start with a triangle. Can you cut it up to make a rectangle?
Age
7 to 14
Challenge level
Problem
Image
Draw a triangle on a piece of paper, and cut it out.
Can you find a way to cut your triangle into no more than four pieces, and reassemble the pieces to make a rectangle?
Is it always possible, no matter what triangle you start with?
If you don't know where to start, click below to see some prompts that may help you:
First prompt:
Start with an isosceles triangle. How could you make it into a rectangle?
Is there a relationship between the base and height of your triangle, and the base and height of the rectangle?
Second prompt:
Draw any triangle, find the midpoints of two of the sides, and join them together. What do you notice?
Third prompt:
Once you have joined the midpoints, can you rearrange the pieces to make a parallelogram? Does that help you to create a rectangle?
Getting Started
You can print out dotty paper from our Printable Resources page if you would like to draw your triangles on a square grid.
The prompts in the problem offer useful suggestions if you are stuck.
The prompts in the problem offer useful suggestions if you are stuck.
Student Solutions
Deana from KIS International School in Thailand/ Taiwan wrote:
First of all draw a right triangle, then cut it in four pieces that are right triangle too. Then it doesn't matter which one you put first because it is the same size, and for me, I made the length of the right triangle 6cm and 8.5cm which that means it is an isosceles triangle. I made 4 tiny triangles and the length is 4.25 and 3 cm.
Letizia from Bangkok Patana in Thailand sent in the following:
To complete this challenging task, I drew an isosceles triangle (with a ruler) on my squared paper to make it easier. Then I cut my triangle out, so I could do the next step. After that, I got my isosceles triangle and cut it in half and then I attached the two pieces back together so now
instead of being an isosceles triangle it was a rectangle.
Here are pictures to show step by step how I completed this process.
These pupils from Broughton Anglican College in Australia sent in their ideas
Adeola
To complete this activity what I did was I found a triangle and ruled then cut it and I then equally halved it then put two sides together to make a rectangle. Remember a rectangle is technically a four-sided shape!
Ebony
The answer is yes because if you get a triangle and cut it in half to make two triangles and put them together it will make a rectangle because two triangles always make a rectangle.
Austin
Cut your triangle into a quadrilateral triangle then cut it right through the middle and then put the to sides back together.
Sam
Cut out a rectangle then gut the rectangle from corner to corner diagonally done.
Laiba from Doha College in Qatar sent in the following excellent work:
There are many different ways to cut and rearrange the pieces of a rectangle into a square. The least amount of pieces that you would need to cut your rectangle into would be two.
If you start with a rectangle with dimensions of the ratio 4:1, you can cut it in half vertically and put one half on top of the other to make a square. This is the only way that you can make a square out of a rectangle by cutting it into only two pieces.
see this clearer here.jpg
EDSR from the Priestlands School in the United Kingdom sent in the following:
These are the triangles I am working with. The lines inside the triangles are the places you should cut
These are the triangles rearranged; as you can see these rectangles used to be triangles:
Answer to Question: Out of all the Triangles I have worked with, you can always make a rectangle by cutting them fewer than four times.
We had four solutions from Norwich Lower School. Firstly, here's what Poppy and George did:
From Else and William:
First, we cut out a big equilateral triangle and drew a rectangle, cut the rectangle out as well as cutting the rectangle in smaller pieces then put it together.
Finally from William and India:
We started with an isosceles triangle and then we split it into two pieces. We then adjusted it and made it into a small rectangle
From a small school in the very north of Scotland - Scourie Primary - we had these four pictures. You can see then enlarged here.jpg.
Their teachers said that although they didn't have time to find a method that would work for every triangle, they decided to split up the task so each experimented with a different idea.
Thank you everyone for sharing your solutions.
First of all draw a right triangle, then cut it in four pieces that are right triangle too. Then it doesn't matter which one you put first because it is the same size, and for me, I made the length of the right triangle 6cm and 8.5cm which that means it is an isosceles triangle. I made 4 tiny triangles and the length is 4.25 and 3 cm.
Letizia from Bangkok Patana in Thailand sent in the following:
To complete this challenging task, I drew an isosceles triangle (with a ruler) on my squared paper to make it easier. Then I cut my triangle out, so I could do the next step. After that, I got my isosceles triangle and cut it in half and then I attached the two pieces back together so now
instead of being an isosceles triangle it was a rectangle.
Here are pictures to show step by step how I completed this process.
Image
These pupils from Broughton Anglican College in Australia sent in their ideas
Adeola
To complete this activity what I did was I found a triangle and ruled then cut it and I then equally halved it then put two sides together to make a rectangle. Remember a rectangle is technically a four-sided shape!
Ebony
The answer is yes because if you get a triangle and cut it in half to make two triangles and put them together it will make a rectangle because two triangles always make a rectangle.
Austin
Cut your triangle into a quadrilateral triangle then cut it right through the middle and then put the to sides back together.
Sam
Cut out a rectangle then gut the rectangle from corner to corner diagonally done.
Laiba from Doha College in Qatar sent in the following excellent work:
There are many different ways to cut and rearrange the pieces of a rectangle into a square. The least amount of pieces that you would need to cut your rectangle into would be two.
If you start with a rectangle with dimensions of the ratio 4:1, you can cut it in half vertically and put one half on top of the other to make a square. This is the only way that you can make a square out of a rectangle by cutting it into only two pieces.
Image
EDSR from the Priestlands School in the United Kingdom sent in the following:
These are the triangles I am working with. The lines inside the triangles are the places you should cut
Image
These are the triangles rearranged; as you can see these rectangles used to be triangles:
Image
Answer to Question: Out of all the Triangles I have worked with, you can always make a rectangle by cutting them fewer than four times.
We had four solutions from Norwich Lower School. Firstly, here's what Poppy and George did:
We found all different triangles the equilateral, the isosceles, the scalene and the right angle. We started with an isosceles and used four small triangles from the big triangles to make a rectangle. The scalene took us quite a lot of time as we tried equal pieces but then realised we needed irregular pieces. We think it is possible as we think we got them
all. The rest were quite similar.
From Tom and Isaac:
We started with an isosceles triangle then we cut it into four pieces then we managed to make it into a rectangle.
From Tom and Isaac:
Image
From Else and William:
Image
Finally from William and India:
We started with an isosceles triangle and then we split it into two pieces. We then adjusted it and made it into a small rectangle
Image
Their teachers said that although they didn't have time to find a method that would work for every triangle, they decided to split up the task so each experimented with a different idea.
Thank you everyone for sharing your solutions.
Teachers' Resources
Why do this problem?
When meeting area and considering methods for working out areas of triangles for the first time, it is tempting to present a formula and invite children to apply it. In this problem, learners are invited to consider that cutting and rearranging a shape does not change its area.Along the way, there are opportunities to practise vocabulary associated with triangles, and apply knowledge of the properties of rectangles in order to generalise.
We hope children will be curious to explore the generality as they wonder whether this can be done with any triangle.
Possible approach
For some students, you may wish to set the problem exactly as it appears in the task:"Draw a triangle on a piece of paper."
"Can you find a way to cut your triangle into no more than four pieces, and reassemble the pieces to make a rectangle?"
Some students might require a little more scaffolding. This could involve providing some templates of particular triangles to begin with, or sharing this prompt from the problem:
Start with an isosceles triangle. How could you make it into a rectangle?
Is there a relationship between the base and height of your triangle, and the base and height of the rectangle?
While students are working, circulate and listen to any discussions they are having. Listen out for key vocabulary and insights about the properties of the shapes they are working with. After most students have created at least one "triangle-rectangle jigsaw", bring the class together to discuss what they have discovered so far. Invite students with useful insights to share them.
Next, introduce the following idea:
"I wonder if it's always possible, no matter what triangle I start with..."
During the next phase of the lesson, you could invite students to share any triangles they have not been able to dissect and rearrange into a rectangle, so that others can have a go.
This could provide a good opportunity to think about angles and side lengths, or a more informal approach considering "long, thin scalene triangles", "short and fat triangles" or other classifications suggested by students.
To develop a convincing argument of the generalisation that all triangles can be cut up and rearranged to make a rectangle, these two prompts from the problem might be useful:
Draw any triangle, find the midpoints of two of the sides, and join them together. What do you notice?
Once you have joined the midpoints, can you rearrange the pieces to make a parallelogram? Does that help you to create a rectangle?
To finish the lesson, bring the class together to share their reasoning and insights. It might be appropriate to leave this as a "simmering task", with space on a "working wall" for students to contribute their thoughts over time. The activity could then be revisited in a future lesson to draw some conclusions.
Key questions
Here are some questions which might tease out students' curiosity while they are working on the task:- Why did you choose that triangle? What else could you try?
- (For a particular example) Is it impossible? Or have you just not found a way to do it yet?
- Are there different methods for different types of triangle?
- Is there a method that works for lots of different triangles?
- To make a rectangle from your triangle, what do you need?
- How do you know that your rearranged shape is a rectangle?
Possible extension
For a challenging extension, students could work on Squaring the Rectangle, working towards a proof that any triangle can be cut up and reassembled to make a square with the same area.
Possible support
Students could draw their triangles on square dotty paper (available on the Printable Resources page), and work out area by counting squares, and then explore possible rectangles with the same area.