I have been given the following equation to use to investigate Isoperimetric Quotients:-

I will begin by investigating the Isoperimetric Quotients of rectangles:-

Using this method I have gathered the following results:-

At this stage I can observe that IQ’s of rectangles are all less than 1. I have marked the shapes that have the same Isoperimetric Quotients as one another with *’s. In each case I observe that the lengths and widths of the shapes are in the same ratio as their partners. To prove this I will investigate 2 more rectangles in ratio with each other:-

This proves that rectangles in the same ratio have the same IQ.

In the table the rectangle with the largest IQ was a Square, and the lowest IQ was a long, thin rectangle. I will now look further into this, and to keep it fair, both shapes will have the same area:-

This is another long, thin rectangle and has a low IQ.

This square has a high IQ.

I can conclude from this that the higher the difference between the lengths and widths of the rectangle, the lower the IQ will be.

Returning to the earlier statement, that rectangles in the same ratio have the same IQ, I can conclude that we can only work out a general formula for squares:-

Summary of rectangles:-

All rectangles in the same ratio have the same IQ. This means we can only find the pattern for regular shapes. ...

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This is another long, thin rectangle and has a low IQ.

This square has a high IQ.

I can conclude from this that the higher the difference between the lengths and widths of the rectangle, the lower the IQ will be.

Returning to the earlier statement, that rectangles in the same ratio have the same IQ, I can conclude that we can only work out a general formula for squares:-

Summary of rectangles:-

All rectangles in the same ratio have the same IQ. This means we can only find the pattern for regular shapes. The larger the difference between the rectangle's length and width, the smaller the IQ will be (when comparing rectangles with the same area). There is a limit as to how high a rectangle's IQ can go. This is the square.

I shall now investigate some other regular shapes, starting with the triangle:-

I will now try another triangle, in the same ratio with this one, to see if the pattern found with rectangles in the same ratio is true for triangles:-

As with the rectangles, if 2 triangles sides are in ratio, they have the same IQ. It now seems clear that because of this pattern, I can only investigate regular shapes, so I shall now investigate equilateral triangles:-

(PTO)

I will now look a little into a general equation for an IQ using triangles:-

I will finish investigating this system later on.

Formula for Equilateral triangles:-

Summary of triangles:-

The largest triangle there can be is the equilateral triangle. This has a lower IQ than the square. This could somehow be connected with the fact it has fewer sides than the square. To investigate this further I shall use pentagons and octagons:-

Formula for regular pentagon:-

The IQ of a regular pentagon is higher than the squares. I will now look at the octagon:-

The octagon has an even higher IQ than the pentagon.

Summary for regular pentagons and octagons:-

The pentagon has a higher IQ than the square, and the octagon has a higher IQ than the pentagon. It seems that as the number of sides a regular shape has the IQ increases. I will study one more shape to prove this:-

Summary for a 20 sided shape:-

The 20 sided shape’s IQ is again bigger than an octagon’s IQ. This proves that as the number of sides a regular shape has increases, it’s IQ increases.

No IQ has yet reached 1. The 20 sided shape was close with 0.993, but not exactly 1. I will now Work out the general formula for any shapes IQ, so I can investigate this apparent limit of 1:-

(PTO)

Now that I have the general formula, I can investigate whether it is possible for a shape to have an IQ of 1 or above:-

The graph on the last page shows how as the number of a regular polygon’s sides increases, the shapes IQ increases. It also shows how no regular polygon exceeds the limit of 1, although as the number of sides increase it always gets infinitely closer to the limit. The only shape that can reach that limit is the circle.

I think this means that an Isoperimetric Quotient is a measurement of how efficiently a shape covers area, with a certain perimeter, or how it covers a certain area with as little perimeter as possible.

I will now investigate this, using a fixed perimeter of 80cm:-

Summary of fixed perimeter investigation:-

The square has the smallest IQ, and also holds the smallest amount of area. The circle has the largest IQ and holds the most area.

CONCLUSION:-

The last investigation proves that IQ must be the measurement of how efficient a shape covers an area with a certain perimeter, or how little perimeter a shape will use to cover a certain area. The worst regular plane polygon is the triangle. There is a limit of 1 that only the circle reaches, and the number of sides a regular polygon has increases infinitely towards that limit. This is probably because as the number of sides increases, the shape becomes more efficient at containing an area.