Investigating the Isoperimetric Quotient of Plane Shapes
Problem:
To investigate the isoperimetric quotient (IQ) of plane shapes using
the calculation shown below.
[IMAGE]
Plan:
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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:
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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
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I will begin by investigating the IQ of equilateral triangles.
Area = 0.5[IMAGE]sin60° = 27.712813
Perimeter = 8+8+8=24
= [IMAGE] = [IMAGE]
= 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 = 0.5[IMAGE][IMAGE]sin60° = 156.31759
Perimeter = 19+19+19=57
=[IMAGE] = [IMAGE]
= 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.
Area = [IMAGE]a[IMAGE]b[IMAGE] sinA
= 0.5[IMAGE]sin60°
=0.5[IMAGE]sin60°
Perimeter = 3[IMAGE]
= [IMAGE] = [IMAGE][IMAGE]
= [IMAGE]
I will now look at the IQ of isosceles triangles.
Area = 0.5[IMAGE]sin80° = 39.884714
* = [IMAGE]- (2[IMAGE]9[IMAGE]cos80°) = [IMAGE]
= 11.570177
Perimeter = 9+9+11.57 = 29.57
[IMAGE] = [IMAGE]
IQ = 0.5733077
The IQ of this isosceles triangle differs from the equilateral
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Surface area of right cylinder (SA) = 2 π r2 + 2 π r h square units