This shows that the greatest area, which can be covered by a RECTANGLE, is 62343.75sq2m. A width of 262.5 and a length of 237.5 obtain this. The greatest area covered by a four-sided shape however, is a square, 250X250, which covers 62,500 sq2m.
Triangles
I am only going to use isosceles triangles. This is because if know the base length, then I can work out the other 2 lengths, because they are the same. If the base is 200m long then I can subtract that from 1000 and divide it by 2. This means that I can say that
Side = (1000 – 200) ÷ 2 = 400.
I will then follow on by investigating the other type of triangle, scalene.
This is how I worked out the area:
b2 = a2 - c2
= 4502 - 502
= 200,000
= 447.2135955
= ½ X 25 X 447.2135955 = 5590.169944m
I produced a graph from this, below:
This again gives good evidence that that a regular sided shape (an equilateral triangle) covers more are than an isosceles triangle, and that scalene triangles cover less area than an isosceles triangle.
As you can see this graph is a-symmetrical. The regular triangle seems to have the largest area out of all the areas but to make sure I am going to find out the area for values just around 333.
This has proved that once again, the regular shape has the largest area. 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 lots of different variables.
Because there are 5 sides, I can divide it up into 5 sectors. Each sector is an isosceles triangle, with the top angle being 720. This is because it is a fifth of 360. This means I can work out both the other angles by subtracting 72 from 180 and dividing the answer by 2 (as it is an isosceles triangle, two of the angles are equal). This gives 540 each. Because every isosceles triangle can be split into 2 equal right-angled triangles, I can work out the area of the triangle, using SOHCAHTOA and ½ base X height. I also know that each side is 200m long, so the base of the triangle is 100m.
Using SOHCAHTOA I can work out that I need to use Tangent.
O 100
T = tan36
This has given me the length of H so I can work out the area.
Area = ½ X b X H = ½ x 100 X 137.638 = 6881.910
I now have the area of half of one of the 5 segments, so I simply multiply that number by 10 and I get the area of the shape
Area = 6881.910 X 10 = 68819.096m2.
I am going to work out the area of the 2 shapes using the same method as before.
Hexagon:
1000 ÷ 6 = 166 1/6 ÷ 2 = 83 1/3.
360 ÷ 6 = 60 ÷ 2 = 30
Area = ½ X b X H = ½ x 83 1/3 X 144.338 = 6014.065
6014.065 X 12 = 72168.784m2
Heptagon:
1000 ÷ 7 = 142.857 ÷ 2 = 71.429
360 ÷ 7 = 51.429 ÷ 2 = 25.714
Area = ½ X b X H = ½ X 71.429 X 148.323 = 5297.260
5297.260 X 14 = 74161.644m2
My prediction was right, so as the number of sides increases, the area increases. Below is a table of the number of sides against area. I have not gone far with this, or produced a graph, as all that proves is the area increase tails off, towards the higher-sided shapes.