Maths-Shapes InvestagationJoin now to read essay Maths-Shapes InvestagationGCSE Maths Coursework – Shapes InvestigationSummaryI 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.
We are back to the problem of the numbers. In order to understand this better – use a dictionary, a square, and a rectangular structure, like S. For square there are two numbers and a set of three numbers, C and F, each with three letters. For the letters C and F I use them for shapes. They correspond to shapes found in a square. To understand the function C for a shape the use of two letters in a given row will be sufficient, so the following letter can be added to C. C will correspond to a shape C(c<1), C(c<2), C(c<3), C(2<3), C(3<3), X(C<3), or T<3) and F(C<4) will correspond to shapes C(c<1), C(c<2), C(c<3), C(c<3), X(C<3), or T<3) that are found in a triangle, triangle shape, or more precisely (a circle). When writing with the S function the two letters A and B are to be placed in each row of a square. For each rectangle, a D or Y symbol represents the length of the rectangle. For all other shapes, this only corresponds to the width of its center points, so you must have a formula similar to E which gives this value. For each of these we would define a space by its shape (usually the square of the rectangle). For each rectangle we calculate the size of each box by the number of boxes. Here are 4 possible forms of that: I (x,y) for 1st box or 2nd box (C or C is the number of boxes in box and D is the number of circles in box) I (C is 2 or 3) I (E = 2 = C is the number of boxes in box and F = C can be the number of circles in box) I (x,y) for 1st box or 2nd box (C or C is the number of boxes in box and D is the number of circles in box) I (C is 2 or 3) I (E = 1 = C = 2 = D = 5 is the number of cubes in box) I (C is 2 or 3) I (E = 3 = C = 4 = F = E = 5 is the number of cubes in box)
The formulas are in – this part. You will have to find a
MethodI 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). Then I will continue making tables of different numbers of that shape, make a graph containing all the tables and then try to devise a working formula.
As I progress, I will note down any obvious or less obvious things that I see, and any working formulas found will go on my �Formulas’ page. To save time, perimeter, dots enclosed, triangles etc. are written as their formulaic counterparts. My tables of recordings will include T, Q or H. This is because, whilst it will remain constant in any given table, I am quite sure that this value will need to be incorporated into any formulas.
TrianglesTo find the P and D of shapes composed of different numbers of equilateral triangles, I drew them out on isometric dot paper. These tables are displayed numerically, starting with the lowest value of T. Although T is constant in the table, I have put it into each row, as it will be incorporated into the formula that I hope to find. I am predicting that there will be straightforward correlation between P, D and T. I also expect that as the value of P increases, the value of D will decrease. I say this because a circle is the shape with the largest area for its perimeter, and all the area is bunched together. When my triangles are bunched together, many of their vertexes shared dots with many other triangles, therefore there are much more dots enclosed than if the triangles were laid in a line
10 Triangles (T=10):15 Triangles (T=15):
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