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

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Introduction

Problem:

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

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

I will begin by investigating the IQ of equilateral triangles.

Area = 0.5sin60° = 27.712813

Perimeter = 8+8+8=24

=     =

= 0.6046

To see if the answer is 0.6046 for all equilateral triangles I will try to find out the I.Q of another equilateral triangle.

Area = 0.5sin60° = 156.31759

Perimeter = 19+19+19=57

=  =

= 0.6046

The answer is the same and has not been affected by the change in the length of the sides. I can now work out a general formula for the IQ of equilateral triangles.

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Area         = ab sinA

        

        =        0.5sin60°

        =0.5sin60°

Perimeter = 3

=   =

=

I will now look at the IQ of isosceles triangles.

Area = 0.5sin80°  = 39.884714

        = - (29cos80°) =  

        = 11.570177

Perimeter = 9+9+11.57 = 29.57

 =

IQ = 0.5733077

The IQ of this isosceles triangle differs from the equilateral triangles which means that it must be the angle of the triangle which affects the IQ as I have already proved that it is not ...

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