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 the lengths of the sides which does so. As the IQ of the equilateral triangle was higher then that of the isosceles triangle it may be more efficient if I spend more time looking at the IQ of regular shapes.
4-sided Polygons
The first four-sided polygon I will be looking at is a rectangle.
Area = 47=28
Perimeter = 4+4+7+7=22
IQ = = = 0.7269801
By looking at how the IQ of a rectangle is calculated I can see that the length of the sides affects the final answer. I can now use this information to find out the algebraic formula for rectangles.
Area = Perimeter =
IQ =
This formula can be adapted to calculate the IQ for squares as shown below:
Area = Perimeter = 4
IQ = = =
Pentagons
cos36 = sin36 =
17cos36= 17sin36=
Area = =13.7532899.9923493
= 137.427675 = 687.13835
Perimeter = 9.992349310 = 99.923493
IQ = = 0.8648062
I can now see that the IQ of shapes is increasing- possibly because the shapes have more sides.
Hexagons
cos30 = sin30 =
8cos30 = 8sin30 =
Area = = 6.9282032
= 27.7128136 = 166.27688
Perimeter = 412 = 48
IQ = = = 0.906899682
Once again I can see an increase in the IQ.
Octagons
cos22.5 = sin22.5 =
9cos22.5 = 9sin22.5=
Area = =8.3149158
=28.637825=229.1026
Perimeter = 3.444150916 = 55.106414
IQ = = = 0.948059448
This seems to confirm the fact that the IQ is affected by the number of sides a shape has.
Decagons
Cos18 = sin18 =
7cos18 = 7sin18 =
Area = = 6.65739562.163119
= 14.40073910 = 144.00739
Perimeter = 2.16311920 = 43.26238
IQ = = = 0.966882799
When looking at the algebraic formula for these shapes to find the general rule the letter will represent the length of the sides.
Algebraic formula for pentagon:
Area=0.5
=0.5
=5 =2.5
Perimeter =
=
= 5
IQ = = =
=0.864806
Algebraic formula for hexagon:
Area = 0.5sin60
=0.5sin60
=5 sin60 =3sin60
Perimeter =
=
= 6
IQ = = =
= 0.906899682
Algebraic formula for octagon:
Area = 0.5sin45
=0.5sin45
=5 sin45 =4sin45
Perimeter =
=
= 8
IQ = = =
=0.948059448
Algebraic formula for decagon:
Area = 0.5sin36
=0.5sin36
=5 sin36 =5sin36
Perimeter =
=
= 8
IQ = = =
= 0.966882799
I can now see that a pattern has emerged a general formula can be used to find the IQ of a shape with sides. This is shown below:
The results of my investigation are shown on the table below
As the number of sides increases it appears that the IQ seems to be increasing. This means that the biggest IQ a shape can have is that of a circle as a circle has an infinite number of sides.
IQ of Circle
= = = 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.