Investigating area and perimeter
Stage 3 – A thinking mathematically targeted teaching opportunity investigating the relationship between area and perimeter when modifying a 2D shape
This investigation is a follow up to Playing with tessellations.
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023
You will need:
rope or string
pencils or coloured markers
ruler or tape measure.
Watch Investigating area and perimeter video (9:52).
(Duration: 9 minutes and 53 seconds)
[White text on a navy-blue background reads ‘Investigating area and perimeter – Playing with tessellations follow up’. In the bottom right corner, the NSW Government red ‘waratah’ logo.]
[Black text on a white background read ‘You will need...’ Bullet points below (read by speaker). On the right, in a still colour image, an orange-handled pair of scissors, a black tape measure, sticky tape, a piece of white string, an orange paper cut-out triangle and a green triangle, and a blue ballpoint pen, and an orange and a green marker pen.]
To complete this investigation we'll need paper, scissors, some sticky tape. You could use either rope or string, some pencils, two different coloured markers and a ruler or some tape measure.
[White text on a blue background reads ‘Let’s investigate!’. In the bottom right corner, a white NSW Government ‘waratah’ logo.]
So mathematicians, let's investigate.
[On a blue and white patterned rug, the green paper cut-out of the triangle.]
After completing the playing with tessellations game, I wondered when we changed the shape of the triangle, what dimensions changed and which ones stayed the same and how would we know? I was interested in finding out whether it changed the area of the triangle and maybe also the perimeter. So, today we're gonna test that out. So, I've created my equilateral triangle using the instructions in the playing with tessellations video, and I actually decided that it would be a good idea to create 2 equilateral triangles.
[The speaker places the orange paper cut-out triangle next to the green one.]
I laid the two different coloured pieces of paper on top of each other and followed the same instructions so that I would have 2 triangles exactly the same. So my first job is to start, just like Sarah did, by changing the shape of my triangle.
[The speaker uses the blue ballpoint pen to draw a line on the triangle (further steps explained).]
I've got my triangle here. I'm gonna draw a squiggly line. It doesn't need to be too dark 'cause I'm gonna cut it out. It just gives me a guideline for my cutting. I'm gonna grab this bit and stick it over the other side.
So, let's take a look now at measuring the perimeter of my two shapes. I thought about using a ruler but because of the curved sides, the ruler isn't the most appropriate tool. So I'm going to use my length of rope. The rope's going to help us to compare the perimeter of the 2 shapes. Now, I know that perimeter is a measurement of length, and it means to measure around the sides of a 2-dimensional shape. So I'm going to measure around here to determine the perimeter of this green triangle first. So I thought I'd start by getting my string. And I've actually got a green marker and an orange marker, one for each shape and I'm going to mark out the perimeter using them on my string so that I can remember… the length of each of the sides. So there's one length, one side and then the other side. Then grab my string and measure, ooh, I missed that corner. Measure the last side. Then I can get my green marker and I'm gonna mark there. I'm gonna look very carefully for the tip of that triangle and I'm gonna mark it on my rope, nice and thick, so I don't lose the perimeter of the green triangle.
So now I'm going to use the same piece of string to measure the perimeter of my orange shape that we made from the triangle to see if they have the same perimeter. So, start by measuring the long straight side. So that's one. Alright, so this one's gonna be a bit trickier. You're gonna have to do a little bit at a time as I curl around these squiggly lines. Keep going around, alright. Oh, how are we looking? So I gotta pull down the rest of my string and keep pulling it round and keeping it on that line. Ooh, and I can see the green mark from when we measured the green triangle coming up and I think the perimeter of this orange shape is actually not going to be the same as the green triangle. That's really interesting. So, there's the final side. I'm going to check like I did before and make sure I can see where I'm marking.
[The white piece of string laid out on the rug. There is a green line and an orange line marked on it.]
So I can see the perimeter now of the green and orange shapes on my length of string here. And I can see that the green triangle's perimeter is not as long as the orange shape that I created out of the green shape. And I think that that's actually really interesting. But what I'd like to do today is find out how much longer is the orange shapes perimeter compared to the green shape. So I'm just going to move that over.
[The speaker uses the measuring tape on the white piece of string.]
Now I can use my measuring tape here to see the difference between the perimeter of the green triangle and the orange shape. And I can see that it's actually, if I go along here, that it's actually 2.5 centimetres longer than the perimeter of the green shape.
I know the orange shape has a longer perimeter, but now I'm interested in seeing if the area was changed when I altered the equilateral triangle. So I know area is the 2-dimensional space inside a region. And one way of comparing the area of 2-dimensional shapes is laying them on top of each other. We call this direct comparison.
[The speaker lays the altered orange triangle on top of the green triangle (further steps explained).]
So here's a triangle that I made at the same time. When I look at it this way, I can see a gap here. There's also an overlap here down the bottom. So what I'm actually now going to do, is I'm gonna cut my shape across the line to see if we can make the orange new shape that we formed to see if it has the same area as the green triangle. So there's my orange triangle laid back on top and I've still got this little bit that I need to try and fit back on. Maybe I need to turn it over. Rotate it and slide it back in there. You can see when I cover it up, and line all the pieces back together that my shapes have the same area.
So I know that my shapes have the same area. By laying them on top of each other using direct comparison, I can see that the orange shape when rearranged covers the whole surface of the green equilateral triangle. The whole space inside the region is covered, so the 2 shapes have the same area.
[White text on a blue background reads ‘What’s (some of) the mathematics?’]
What's some of the mathematics?
[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker). Below, 2 small colour images of the green and orange triangles side by side and then on top of one another.’
When we are measuring area and perimeter we are measuring two different things. When we're measuring the size of a surface, we're measuring the area. We can measure this in different ways, like directly comparing. Today we directly compared by laying the cut up surfaces of our curvy-edged ex-triangle over the top of the other surface of our green triangle to determine if the areas were bigger, smaller or if they were the same.
[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker). Below, 2 small colour images of the orange ex-triangle and the green triangle being measure by the white string.’]
When we measure the length around the outside of the shape, we're working out its perimeter. Did you know this cool fact? The word perimeter comes from the Greek word perimetros, where metron means 'measure' and peri means 'around'. We can determine the perimeter in different ways. We knew we couldn't use a ruler for our new curvy-edged ex-triangle, so we used rope. Once we'd wrapped the rope around the outside of the shape, we could lay it out straight to compare the perimeters.
[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker). Below, on the left, 2 small colour images of the orange ex-triangle and the green triangle being measure by the white string. Black text above reads ‘Perimeter’. On the right, 2 small colour images of the 2 triangles side by side and on top of each other. Black text above reads ‘Area’.]
When we changed the shape from being a triangle to a curvy-edged ex-triangle, we noticed the area stayed the same but the perimeter changed. So now we know a shape can have the same area but the perimeter can be different. So, over to you now mathematicians. Can you find some shapes that look different but have the same area? What do you notice about the perimeter of the shapes that you found?
[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]
[End of transcript]
Draw and cut out 2 equilateral triangles.
Draw a squiggly line from one corner to another. Cut along the squiggly line.
Stick the part you cut off on another side using sticky tape to create a new shape.
Measure the perimeter of the triangle by guiding the rope around the three edges of the triangle.
Make a mark on the rope to show the length of the perimeter.
Measure the perimeter of the new curvy ex-triangle by guiding the rope around the edges of the shape. Make a mark on the rope to show the length of its perimeter.
Use a ruler or measuring tape to explore the difference.
Is the perimeter of your shapes the same or different?
If one shape’s perimeter is longer than the other, how much longer is it?
Is the area of your shape the same or is it different?
Cut your shape back up in its pieces and use direct comparison to see if the area changed.
Can you change a shape so that the area and perimeter stay the same?
Find some other shapes to explore. What would happen if we created a new shape out of a square, hexagon or a different type of triangle would the perimeter change or would it stay the same?
Share your work with your class on your digital platform. You may like to:
share pictures of your work
comment on the work of others.