Investigating different triangles
I am first going to investigate a few scalene triangles with a perimeter of 1000m. I drew the triangles by hand using a scale of 1cm for 100m. Then I measured the angle at the bottom right. To find the area I used the formula: ½absinθ
As you can see the areas of the scalene triangles are not as much as the square, which had an area of 62500m2. Now I am going to investigate isosceles triangles in a table.
I entered the bases of 0m and 500m in the table. I know these measurements do not make a triangle but they are needed for the graph to be complete. In the table the areas of the triangles go up and then down. This means when the difference between the sides is less, the area is larger. This graph is not symmetrical because the maximum is not in the middle.
The largest area is 48112.5224252520m2. This is when all the sides are 333/1/3 m (equilateral triangle). To check that it is the maximum area, I inserted values just before it and after it. The areas of those values were less, so 48112.5224252520m2 is the largest area for a triangle of perimeter 1000m. This proves my theory in the beginning that regular polygons will give the largest area.
Investigating Pentagons
I am first going to look at an irregular pentagon, the house pentagon.
Now I am going to look at the regular pentagon of all sides = 200m.
So as you see the regular pentagon is bigger than the irregular pentagon. The area of the pentagon is 68819.09602m2. This is the largest area so far in the investigation. I am now convinced that regular shapes have the largest area. Another pattern that is noticed is that as you increase the amount of sides in regular shapes (1000m perimeter) the areas increase.
Investigating regular hexagons
Now I am going to have a look at the regular hexagon of 1000m perimeter.
These calculations backup my predictions I made earlier. I will only look at regular shapes from now on because they give the largest area.
Investigating Polygons with n sides
Now that I have found out that regular shapes give the largest area, I am going to investigate regular polygons with an increasing number of sides. To do this I am going to draw out a table and a graph showing the relationship between the number of sides and the area. Below is the formula to find the area of the polygon. The diagram shows the triangle in the polygon.
When trying to figure out the height, I had to calculate TAN (180/n). In excel I could not use 180o because excel does not process degrees. Instead I used radians. I found out from Encarta Encyclopedia, that one radian is a unit of angle, equal to the central angle inscribed in a circle the arc of which is equal in length to the radius of the circle. So if the circumference of a circle is 2Pir and the angle is 360o, 2PI radians is equal to 360o (r subtends one radian). So 180o is equal to PI.
This graph is not a continuous graph and the data is discreet because you cannot have an area for 3/1/2 sides. I have kept the line though so I can show how it evens out.
The values in the table are correct. I checked the 5 and 6 sided shape with the answers I got for the pentagons and the hexagons.
From the table, there are several patterns. As the number of sides increased the base, the angle at the top of the triangle and area of each triangle get smaller. Also as it increased, the height and area of the polygon increased.
Also as you increase the sides, the area is tending towards a certain number. This can be seen on the graph. To find that certain number I am going to draw the same table with a larger amount of sides on the next page.
As the number of sides increases, the area tends to the area of the circle which is 79577.4715459451m2, This is also shown on the column, which is the difference between the area of the circle and the area of the polygon. This decreases as n increases. The area of the circle and the area of the n-sided polygon when n tends to infinity are almost the same because a circle has an infinite number of sides. So as the number of sides for the polygon tends towards infinity, the area becomes more like the area of a circle.
Although even if you keep on increasing the number of sides in the polygon it will never have the exact area as the circle. This is because the graph the polygon forms is an asymptote. It will never touch the value, which is the area of the circle. Also the circle has an infinite number of sides and you can not write infinity down as a number, so there is no way of calculating the polygon with infinite number of sides. So the circle will always have the largest area of perimeter 1000m.
To explain this I will look at the table. As the sides get bigger, the base gets smaller, the angle at the top gets smaller and the height gets bigger and the triangle becomes more and more like a straight line, which is like the radius of the circle (159.1549431m). This is also shown on the column on the table, which is the difference between the height and the radius. The difference gets smaller as n increases. To explain this even better, I have researched an A-level book to provide rigorous proof.
In the A-Level book, it says applying Tan to a very small angle is the same as the angle itself. So when n tends to infinity and PI/n becomes very small, Tan (PI/n) would be the same as PI/n. To test this we can have a look at the table. As n gets larger, the column PI/n becomes the same as Tan (PI/n). After finding this out, I can compare the formulas to find the area of a circle with the one to find area of an n-sided shape when n tends to infinity and try to find the connection. I have used radians because the area for the circle has PI in it and the area of the polygons has 180o in it. So I can change the 180o into PI so it is easier to compare the formulas.
So as you can see the formulas for the height of the triangle in the polygon when n tends to infinity and the radius of the circle are equal. Also the formulas for the area of the circle and the area of the polygon when n tends to infinity are equal. That is why the values are almost equal in the table.
Conclusion
I conclude that the largest shape of 1000m perimeter the farmer can use is a circle of radius 159.1549431m that gives an area of 79577.4715459451m2.
Malcolm Manekshaw
Dubai College
