Essay: Patterns formed by increasing the scale factor – maths

MATHS COURSEWORK-1
Question: – To investigate the patterns formed in perimeters and areas of different triangles/polygons by increasing the scale factor.
Introduction-
Corresponding sides change by the same scale factor. What does this mean? It means that all the sides of the small figure are multiplied by the same number to obtain the lengths of the corresponding sides of the large figure.
Perimeter Examples
Let’s talk about how scale factors influence shapes in geometry, starting with perimeter.
Here’s a rectangle. Its width is 3 and its length is 5. What’s the perimeter? 3 + 3 + 5 + 5, or 16. Now let’s apply a scale factor of 4 for the new rectangle. It will have dimensions that are 4 times that of the original. Instead of a width of 3, it will be 3 x 4, or 12. And instead of a length of 5, it’ll be 5 x 4, or 20. This new rectangle is similar to the original, which means it has the same shape, but not necessarily the same size.
What impact did this have on the perimeter? The new perimeter is 12 + 12 + 20 + 20, or 64. 16 to 64? That’s 4 times the original. So the change in perimeter is equal to the scale factor.
Area Examples
Things are a little different with area. Let’s look at some two rectangles. The area of a rectangle is length time’s width, so it’s 5 x 3, which is 15. What about the second one? 20 x 12, which is 240. Does 15 x 4 = 240? No. What is the relationship between 15 and 240? If you divide 240 by 15, you get 16. And what was our scale factor? 4. 16 is 4^2. So the change in area is equal to the scale factor squared.
Let’s look at another example. Here’s a triangle with a base of 5 and a height of 4. The area of a triangle is ½ times base time’s height. So this triangle’s area is ½ x 5 x 4, which is 10. Let’s make a new triangle using a scale factor of 3. This new triangle has a base of 15 and a height of 12. Its area is ½ x 15 x 12, or 90.
Okay, remember that our scale factor was 3. That’s 9. So the change in area should be 9 times the original. Does 10 x 9 = 90?
When you’re working with scale factors, square the scale factor to determine the area of the new figure. If you think about it, it makes sense why area would be the scale factor squared. Area involves two dimensions multiplied together. With scale factor, all you’re really doing is multiplying the scale factor times itself.
• The main purpose of this activity is to investigate the pattern formed, when the scale factor is increased. To know how the sizes of different polygons depends on its scale factor. To investigate the relationship of the sides of similar figures by looking at patterns formed by the ratios. To know more information about scale factors. This is the main purpose doing this coursework.
• The main objective in this coursework is to know how to find the scale factor of a given polygon. To find out the result deriving from increasing the scale factor of various triangles/polygons. Compare the shapes resulted in the increase of scale factor. To find the proportion of the increase in the size to the ratio. These are the main objectives in this coursework.
• The description of maths in this coursework is that we are using the scale factor and comparing the sides and the various polygons formed. We will be comparing the different measurement each side of a polygon has obtained to the others which will be formed. This is how maths is being described in this coursework
• The mathematical content in this coursework is that we are using various formulas to find the area/perimeter of different polygons. Sides are used as a mathematical content. We are using a lot of maths in this coursework. We will be using various mathematical concepts in finding the area and perimeter of the various polygons by increasing its scale factor. This is the mathematical concept used in this coursework.
What is a polygon?
Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is “closed” (all the lines connect up).
A polygon is any 2-dimensional shape formed with straight lines. Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. The name tells you how many sides the shape has. For example, a triangle has three sides, and a quadrilateral has four sides. So, any shape that can be drawn by connecting three straight lines is called a triangle, and any shape that can be drawn by connecting four straight lines is called a quadrilateral.
Polygon
(straight sides) Not a Polygon
(has a curve) Not a Polygon
(open, not closed)
Polygon comes from Greek. Poly- means “many” and -gon means “angle”.
Types of Polygons
Regular or Irregular
A regular polygon has all angles equal and all sides equal, otherwise it is irregular
Regular Irregular
The pictures shown above are some examples of some of the polygons. As seen they are all closed figures and are called polygons.
Each shape has a different area and perimeter. So as the scale factor increase even the area and perimeter would obviously increase.
2D SHAPES-
Let us look at the area of some polygons-
Area is the measurement of a figure or a shape or a size of a surface.
Perimeter of a shape is the boundary/distance around its edge.
Let us look at the perimeter of some polygons-
SCALE FACTOR-
A scale factor is a number which scales, or multiplies, some quantity. In the equation y = Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x.
In two similar geometric figures, the ratio of their corresponding sides is called the scale factor. To find the scale factor, locate two corresponding sides, one on each figure. Write the ratio of one length to the other to find the scale factor from one figure to the other.
For example we will take a simple example, in the figure given below.
In this figure the scale factor is 2.
The sides are proportional and equal in this figure.
The figure is getting enlarged because of having a scale factor of 2. A figure may only not just enlarge even it can diminish or get compressed.
As seen in the table above the sides of each shape is different is obvious. In most shapes a part of the name denotes the number of sides. For example Octagon, Oct meaning Eight.
Focusing on 2D shapes-
SQUARE
Case-1, scale factor-2
For example if we are taking a square of side 2 cm and enlarge it by a scale factor of 2 the new side would be 4 and enlarge that it would be 8. Since the scale factor is 2, C=2
Original length New length (y=CX)
2 4
4 8
8 16
16 32
Now, let’s look at the areas of the squares.
Length (s) Area= s2
2 4
4 16
8 64
16 256
Now, let’s look at the perimeters of the squares.
Length (l) Perimeter=4l
2 8
4 16
8 32
16 64
Case-2, scale factor-3
Original length New length (y=CX)
2 6
6 18
18 54
54 162
Now, let’s look at the areas of the squares.
Length (S) Area=s2
2 4
6 36
18 324
54 2916
Now, let’s look at the perimeters of the squares.
Length (l) Perimeter=4l
2 8
6 24
18 72
54 216
Comparing the two cases-
In these both cases we are taking the scale factor as 2 and 3. The Square is a shape with all its sides equal and even the new length varies.
Rectangle
Case-1, scale factor 2
We have a rectangle with a side of 2 and 3. If we enlarge the rectangle with a scale factor of 2, the new sides will be 4 and 6 and so on. Let’s look at all the values. c = 2
Original length Original Breadth New breadth and length
3 2 4,6
6 4 8,12
12 8 16,24
24 16 32,48
Now, let’s look the areas of the new rectangle.
Lengths Area= b x l
2,3 2 x 3 = 6
4,6 4 x 6 = 24
8,12 8 x 12 = 96
16,24 16 x 24 = 384
Now let’s look at the perimeter of the new rectangle.
Lengths (s) Perimeter = 2(l+b)
2,3 2(2+3) = 10
4,6 2(4+6) = 20
8,12 2(8+12) = 40
16,24 2(16+24) = 80
Case-2, scale factor-4
We have a rectangle with a side of 2 and 3. If we enlarge the rectangle with a scale factor of 4, the new sides will be 4 and 6 and so on. Let’s look at all the values. c = 4
Original length Original Breadth New breadth and length
3 2 12,8
12 8 48,32
48 32 192,128
192 128 768,512
Now, let’s look the areas of the new rectangle.
Lengths Area= b x l
2,3 2 x 3 = 6
8,12 8 x 12 = 96
32,48 32 x 48=1536
128,192 128 x 192=24576
Now let’s look at the perimeter of the new rectangle.
Lengths (s) Perimeter = 2(l+b)
2,3 2(2+3) = 10
8,12 2(8+12) = 40
32,48 2(32+48)=160
128,192 2(128+192)=640
Comparing the two cases-
As we are comparing the two cases in a rectangle we do know that the opposite sides are equal but none the less the area and perimeter do differ.
3D SHAPES-
Only the 3D shape or object itself can occupy its own space. For example, no other human can stand where you are standing. In mathematics, there are standard 3D shapes such as spheres, cubes, prisms, cones, and pyramids.
An object that has height, width and depth, like any object in the real world. Example: your body is three-dimensional. Also known as “3D”.
The volume of a 3D shape can be expressed as area of cross section x height and surface area can be expressed as the sum of the area of its faces but, it may vary for different 3D shapes.
Now, let us take a look at the formula for various 3D shapes.
Cube:
Case 1: scale factor is 4
We have a cube with side 4 and if we enlarge it with a factor of 4 the new side will be 16 and so on. Let’s look at the new lengths.
Original length (x) New length (y =cx)
4 16
16 64
64 256
256 1024
Now the new volumes and surface area will be:
Side Volume s3
4 64
16 4096
64 262144
256 16777216
1024 1073741824
Now the new surface area will be:
Side Surface area 6xs2
4 96
16 1536
64 24576
256 262144
1024 4194304
Case 2: the scale factor is 3
Original length (x) New length (y =cx)
4 12
12 36
36 108
108 324
324 972
972 2916
Now the new volumes and surface area will be:
Side Volume s3
4 64
12 1728
36 46656
108 1259712
324 34012224
Now the new surface area will be:
Side Surface area 6xs2
4 96
12 864
36 7776
108 69984
324 629856
Comparing the two cases-
Now for a cube all sides are equal, we are taking scale factors of 3 and 4 and are enlarging the 3D shape; the volume and the surface area are different in all these cases.
Cuboid:
Case 1: scale factor is 2
We have a cuboid with sides of 4, 5 and 6. If we enlarge the cuboid with a scale factor of 2, the new sides will be 8, 10 and 12 and so on. Let’s look at all the values. c = 2
Original Breadth Original length Original height New breadth and length and height
4 5 6 8, 10, 12
8 10 12 16, 20, 24
16 20 24 32, 40, 48
32 40 48 64, 80, 96
64 80 96 128, 160, 192
Now their volumes and surface areas will be:
Sides Volume lbh Surface area 2(lb+bh+hl) Position
4, 5, 6 120 60 1
8, 10, 12 960 120 2
16, 20, 24 7680 240 3
32, 40, 48 61440 480 4
64, 80, 96 491520 960 5
Case 2: scale factor is 10
Original Breadth Original length Original height New breadth and length and height
4 5 6 40, 50, 60
40 50 60 400, 500, 600
400 500 600 4000, 5000, 6000
4000 5000 6000 40000, 50000, 60000
40000 50000 60000 400000, 500000, 600000
Now their volumes and surface areas will be:
Sides Volume lbh Surface area 2(lb+bh+hl)
4, 5, 6 120 60
40, 50, 60 120000 600
400, 500, 600 120000000 6000
4000, 5000, 6000 120000000000 60000
40000, 50000, 60000 120000000000000 600000
Comparing the Two cases:
In a cuboid, in two cases we are using scale factors 3 and 10 and enlarging the 3D shape, the volume and surface area is different.
The following are the microworlds for this coursework-
Conclusion-
We conclude by saying that the area and the perimeter of a polygon depend on the scale factor. We also did prove the same in this coursework, as the scale factor was we did compare what happened in each case.
THANK YOU
By- P.Maniesh Raj
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