This is also a parabola, therefore a square is the rectangle with a larger area, I can try this with other regular polygons
I will try isosceles triangles because I can vary the base easily with this, I believe that the biggest area will belong to the equilateral triangle
The formula for working the isosceles triangle out is simple. Split the base in half giving two right angles. Use Pythagoras to work out the triangles height using the base length divided by two and the other side length. Multiply height by base then divide by two for the area.
See Graph 2
To represent the Equilateral triangle, notice it is the largest
In this case it’s not a parabola, this being different again I will check that 333.3 recurring is the biggest area.
It is the largest of the isosceles triangles, this could be a pattern for all regular shapes. I now believe that if the circle has the largest area (not yet proved) then the more sides, the closer to the circle, the bigger area, there is little difference between a trillion sided polygon and a circle
Because the last 2 shapes have had the largest areas when they are regular, I am going to use regular shapes from now on. This would also be a lot easier as many of the other shapes have millions of different variables.
The next step is to investigate the regular polygons. I need a formula to work out the area of a regular polygon by just looking at its number of sides, So I look at a pentagon
If I use trigonometry I can work out isosceles triangles radiating out from the center. As there are 5 sides, that makes 5 triangles. The top angle is 720 (360/5). I can work out both the other angles by subtracting 72 from 180 and dividing the answer by 2. This gives 540 each as the other angles. Because every isosceles triangle can be split into 2 equal right-angled triangles, I can work out the area of the triangle, using trigonometry. I also know that each side is 200m long so;
Tangent makes 137.638 as the height
This has given me the height so I can work out the area of the isosceles triangle
(Height X Base) / 2 = 13763.8192
I can now just multiply that number by 5 and I get the area of the shape
Area = 13763.8192 X 5 = 68819.096m2.
My predictions were correct and as the number of side’s increases, the area increases. Below is a table of the number of sides against area
See Graph 3
You notice in the graph that the area change is great to begin with but decreases as you get further on so the curve gets slowly flatter and flatter, but never is actually flat, there is another column showing change in
From the method that I used to find the area for the pentagon, hexagon and heptagon I ca work out a formula using n as the number of sides. To find the length of the base of a segment I would divide 1000 by the number of sides, so I could put , but as I need to find half of that value I need to put . All
From the first graph, the line straightens out as the number of side’s increases. Because I am increasing the number of sides by large amounts and they are not changing much, this indicates a graph of this would continue this trend.
I am going to see what the result is for a circle. Circles have an infinite amount of sides, so I cannot find the area by using the equation that I have used to find the other amount of sides out. I can find out the area of a circle by using pi.
- 1000/pi = diameter
- diameter / 2 = radius
- pi X radius squared = area
-
which for a 1000m circumference 79577.47155 metres squared
This is larger than any number of sides and would fit on the graph, but this difference can become almost imperceptible.
Therefore, I have proved and conclude that the polygon which will give the largest area, for a perimeter of 1000 metres is the circle. It is also quite safe to conclude that this is the same for any perimeter because the size of perimeter and scale in theory make no difference.
