Sides: 299m, 299m, 301m, 301m
299m x 301m = 89999m2
Area: 89999m2
The results from this calculation prove my initial findings, that the square had the largest area. Though the result from the above calculation does tell me that quadrilaterals close to being squares are have larger areas.
Triangles
Now I will investigate the triangles, having concluded that the square provides the largest area from all the shapes that I have investigated so far.
To find the areas of isosceles and equilateral triangles I will split them in half to make two right-angled triangles. I will use Pythagoras’ theorem to find the heights of the triangles, then the formula:
Area= (Base x Height)/2 to find the areas.
Equilateral triangle
1. Base: 400m Other Sides: 400m, 400m
h = 4002 - 2002 = 346m (Pythagoras’ theorem)
400 x 346 = 69200m2
2
Area: 69200m2
Isosceles triangles
1. Base: 200m Other Sides: 500m, 500m
h = 5002 – 1002 = 490m
200 x 490 = 49000m2
2
Area: 49,000m2
2. Base: 500m Other Sides: 350m, 350m
h = 3502 – 2502 = 245m
500 x 245 = 61250m2
2
Area: 61250m2
Right-angled triangle
5. Base: 300m Other Sides: 400m, 500m
300 x 400 = 60000m2
2
Area: 60000m2
This table shows the areas of the different triangles:
This shows that the equilateral triangle has the largest area. To check that this was true, I found the area of a triangle whose sides were close to the length of the sides of the equilateral triangle.
Base: 401m Other Sides: 399m, 400m
s (s – 401) (s – 399) (s – 400) = 69281m2
Area: 69281m2
From my calculations I have decided that the equilateral triangle is the triangle, which will provide the greatest area with a 1200m perimeter.
Polygons
The next group of shapes that I will investigate are the polygons, there can be many, many different shaped polygons, with many sides, so I will make a chart using excel to calculate the areas of polygons with a great number of sides.
I order to find the areas of polygons I will have to split them up into triangles. Each polygon is made from a large number of triangles, which is equal to the number of sides in the polygon. The formula I will use to calculate the polygon’s area will be:
360000/(n tan (180/n))
To simplify this I will find the number of triangles in the shape, this figure is equal to the number of sides in the polygon (n). So, I will find the area of one triangle and multiply it by n to find the area of the polygon. I will find the length of the base of the triangle by dividing 1200m by n, and I will find the angle opposite the base by dividing 360o by n. I will then use trigonometry to find the height of the triangle, and the formula
Base x Height to find its area, before I multiply it by n.
Pentagons
1. Sides: All 240m
360o = 72o 1200m = 240m
5 5
tan 36 = 120
h
h = 120 = 165m
tan 36
240 x 165 = 19800m2
2
5 (19800m2) = 99000m2
Area: 99000m2
Octagon
2. Sides: All 150m
360o = 45o 1200m = 150m
8 8
tan 22.5 = 75
h
h = 75 = 181m
tan 22.5
150 x 181 = 13575m2
2
8 (13575m2) = 108600m2
Area: 108,600m2
My theory is that it is possible to tell the area of other polygons with a much greater amount of sides using the same formulae.
Now though I am beginning to notice a trend in the relationship between the number of sides on a shape and the area, it would appear that both rise with each other. On the following spreadsheet my calculations are much more accurate, and the angles are measured in radians.
The definition of a radian according to Microsoft is “Syntax
Radians (angle)
Angle is an angle in degrees that you want to convert.
Example: RADIANS(270) equals 4.712389 (3π/2 radians)”
Circle
A Circle has no sides or an infinite number of sides. So, if my theory is correct the circle should provide the largest area from all the shapes I have investigated.
Circumference: 1200m
Diameter = 1200 = 382m
Π
Radius = 382 =191m
2
Area = Π x 1912 = 114608m2
Area: 114608m2
.
114608m2 is the largest area that I have achieved using a perimeter/circumference of 1200m. This means that the circle would be the most effective shape to use in ‘The Fencing Problem’.
Conclusion
From the results I have gathered throughout my investigation I have come to the conclusion that the circle provides the largest surface area. In this investigation I tested quadrilaterals, triangles, polygons and a circle. Having tested all these shapes I came to the conclusion that the greater the amount of sides the greater the area.
The circle has the largest area because it has an infinite amount of sides. With 1200m of fence you can enclose a circle with a 1200m perimeter and 114608m2 area, this is the greatest area possible to enclose with that amount of fence.
