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To investigate the isoperimetric quotient (IQ) of plane shapes using the calculation shown below.

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Introduction

Introduction

Problem:

To investigate the isoperimetric quotient (IQ) of plane shapes using the calculation shown below.

image00.png

Plan:

I will start off by investigating three sided shapes and then increasing the number of sides as I go along. I will be looking at how different factors such as the number of sides on a shape and the length and angle of the sides of the shapes affect the isoperimetric quotient.

Hypothesis:

I have come up with a number of predictions as to what I think the outcome of my investigation will be:

  1. As the number of sides on a shape will increase so will the IQ.
  2. The length of the sides of any given shape will not affect the IQ of the shape.
  3. The angle of the sides of any given shape will also have affect on the IQ of the shape.

Triangles

...read more.

Middle

9.9923493

                = 137.42767image02.png5 = 687.13835

Perimeter = 9.9923493image02.png10 = 99.923493

IQ = image54.png  = 0.8648062

I can now see that the IQ of shapes is increasing- possibly because the shapes have more sides.

Hexagons

cos30 = image55.png                sin30 = image56.png

8cos30 = image51.png                8sin30 = image52.png

Area = image53.png        = 6.9282032image58.png

                = 27.712813image02.png6 = 166.27688

Perimeter = 4image02.png12 = 48

IQ = image59.png          = image60.png  = 0.906899682

Once again I can see an increase in the IQ.

Octagons

cos22.5 = image55.png                sin22.5 =image61.png

9cos22.5 =image51.png                9sin22.5= image52.png

Area = image53.png        =8.3149158image63.png

                =28.637825image64.png=229.1026

Perimeter = 3.4441509image02.png16 = 55.106414

IQ = image65.png = image66.png = 0.948059448

This seems to confirm the fact that the IQ is affected by the number of sides a shape has.

Decagons

Cos18 = image67.png                sin18 =image68.png

7cos18 = image51.png                7sin18 =image52.png

Area = image51.pngimage70.png        = 6.6573956image02.png2.163119

                = 14.400739image02.png10 = 144.00739

Perimeter = 2.163119image02.png20 = 43.26238

IQ = image71.png         = image72.png = 0.966882799

When looking at the algebraic formula

...read more.

Conclusion

IQ of Circle

=image25.png        = image26.png  = 1

This means that the maximum IQ of a shape must be 1.        

Conclusion

My original problem was to investigate the IQ of plane shapes. I started off by looking at shapes with 3 sides (triangles) and finished by looking at the IQ of circles. I decided to look at regular polygons (with the exception of isosceles triangles) to make it easier to calculate the IQs of shapes. When looking at rectangles I discovered that the angle of the shape wasn’t always essential to what the IQ turn out to be. After my investigation of regular polygons I discovered that the first part of my hypothesis was correct- as the shape gained more sides the IQ of the shape increased.

Looking at a scatter diagram displaying the results of the investigation can show this.


image27.png

...read more.

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