# Shapes made up of other shapes

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Introduction

GCSE Maths Coursework - Shapes Investigation

Summary

I am doing an investigation to look at shapes made up of other shapes (starting with triangles, then going on squares and hexagons. I will try to find the relationship between the perimeter (in cm), dots enclosed and the amount of shapes (i.e. triangles etc.) used to make a shape.

From this, I will try to find a formula linking P (perimeter), D (dots enclosed) and T (number of triangles used to make a shape). Later on in this investigation T will be substituted for Q (squares) and H (hexagons) used to make a shape. Other letters used in my formulas and equations are X (T, Q or H), and Y (the number of sides a shape has). I have decided not to use S for squares, as it is possible it could be mistaken for 5, when put into a formula. After this, I will try to find a formula that links the number of shapes, P and D that will work with any tessellating shape – my ‘universal’ formula. I anticipate that for this to work I will have to include that number of sides of the shapes I use in my formula.

Method

I will first draw out all possible shapes using, for example, 16 triangles, avoiding drawing those shapes with the same properties of T, P and D, as this is pointless (i.e. those arranged in the same way but say, on their side. I will attach these drawings to the front of each section. From this, I will make a list of all possible combinations of P, D and T (or later Q and H).

Middle

22

0

10

13 Squares (Q=10):

P=

D=

T=

16

6

13

18

5

13

20

4

13

22

3

13

24

2

13

26

1

13

28

0

13

16 Squares (Q=16):

P=

D=

T=

16

9

16

18

8

16

20

7

16

22

6

16

24

5

16

26

4

16

28

3

16

30

2

16

32

1

16

34

0

16

Firstly I will test my previous formulas, P=T+2-2D, D=(T+2-P)/2 and T= P+2D-2, to see if they hold true – of course, substituting T with Q. If the formulas still hold true, I will be able to save lots of time trying to find a formula linking P, D and Q. Even if they don’t, all will not be lost – all the answers may be incorrect by, say, 4. Therefore I could make slight modifications (i.e. +4) to the existing formulas to get a new, working formula for squares.

So where P=14, D=4 and Q=10…

P=10+2-8 Ð P=4 DP is out by –10, or -Q

D=(10+2-14)/2Ð D=-1 DD is out by –5

Q=14+8-2 Ð Q=20 DQ is out by +10, or +Q

And where P=16, D=6 and Q=13…

P=13+2-12 Ð P=3 DP is out by –13, or -Q

D=(13+2-16)/2Ð D=-0.5 DD is out –6.5

Q=16+12-2 Ð Q=26 D Q is out by +13, or +Q

And where P=16, D=9 and Q=16…

P=16+2-18 Ð P=0 D P is out by –16, or -Q

D=(16+2-16)/2Ð D=1 D D is out by –8

Q=16+18-2 Ð Q=32 DQ is out by +16, or +Q

From these trials it is clear that the formulas for triangles doesn’t work properly, but there is some correlation which could help me to find a formula for squares. For example, the formulas always give a value of Q to be twice its real value, and the value of P is always less than its real value by whatever Q is.

So, in theory, all I need to do to get a working formula for Q= is to change the formula from Q=P+2D-2 to Q=(P+2D-2)/2, or more simply Q=P/2+D-1.

Conclusion

I also successfully found a formula that will give you the maximum perimeter of a shape, so long as you know what shape it is and how many there are.

Evaluation

I would say that this investigation has been a success. I managed to find a link between P, D and T, which progressed into a link between P, D and Q or H, then on to my universal formulas. After that I went a stage further and developed a formula that would tell you the maximum possible perimeter of a shape.

Unfortunately, my investigation was hindered by 2 things – foremost was my lack of time to carry out the investigation as far as possible (i.e. researching 3D shapes, and irregular tessellating shapes, such as ‘L’ and ‘T’ shapes), and second I did not have any dotted paper capable of drawing regular pentagons or heptagons, as I could have looked at these along with my triangles, squares and hexagons.

If I were to redo this investigation, I would make sure that I set aside enough time to do a proper job of it, and looking at my own made-up shapes to see if my formulas still applied. I also would have very much liked to have been able to move on to looking at 3D shapes, but doing this means wither physically making paper or card squares / pyramids etc, which is vastly time-consuming, or using multi-cubes, of which I have none.

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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