The results are as followed:
Using this results table I can draw another graph of area against base of triangle. From the graph, I can see that the maximum area of a isosceles triangle with a range of bases, ranging from 320m to 340min in 0.5m steps, is a triangle with a base of 333.5m, with sides of 333.25m long and a perimeter of 1000m. The area of this triangle is 48112.504m². To work out the area of the triangle more accurately, I will change the base of the triangle in smaller steps of 0.1m.
The results are as followed:
Using this results table I can draw a final graph of area against base of triangle. From the graph, I can see that the maximum area of a triangle with a range of bases, ranging from 333m to 334m, is a triangle with a base of 333.3m, with sides of 333.3m long and a perimeter of 1000m.The area of this triangle is 48112.522m². The triangle is an equilateral triangle.
Quadrilaterals
I will investigate if and how the interior angle of a parallelogram effects the area of the parallelogram and see if the parallelogram has a bigger area than a rectangle with the same perimeter of 1000m. To work out the area of a parallelogram, I will use the following formula:
l = length of parallelogram
w = width of parallelogram
h = perpendicular height of parallelogram
a = area of parallelogram
a = l * h
However, because I only know the length and width of the parallelograms, I will need to work out the perpendicular height. For example, if the length of a parallelogram was 300m and the width of a parallelogram was 200m, I wouldn't be able to use the formula shown above. The formula to work out the perpendicular height is:
h = w * sinx°
I used sine because I know the hypotenuse (width) and I want to work out the opposite (perpendicular height). I will now work out the area of the following parallelogram.
h = 200m * sin45°
= 141.4m
a = 300m * 141.4m
= 42420m²
I will change the angle x but keep the length and width the same to find the interior angle needed to create the maximum area. The results were as followed:
These results show that a parallelogram has the greatest area when angle x is 90° i.e. when it is a rectangle. Therefore, a rectangle has a greater area than a parallelogram.
I will investigate the areas of different rectangles, that each have the same perimeter of 1000 metres, because a rectangle is an easier shape for which to work out the area for than any other quadrilaterals. To work out the area, I will use the following formula:
l = length of rectangle
w = width of rectangle
a = area of rectangle
a = l * w
The two examples I worked out are shown below.
The area of the rectangles above are as follows.
- area = length * width
area = 150m * 350m
area = 525000m²
- area = length * width
area = 200m * 300m
area = 60000m²
The examples above show that not all rectangles with the same perimeter have the same area.
The formula to work out the area of any rectangle, with a perimeter of 1000m for any given length is:
width = 500 - length
area = length * width
Substituting the width in the formulas above creates this formula:
area = length * (500 - length)
area = (500 * length) - length²
I'm going to use the formula shown above to find the maximum area of a rectangle with a perimeter of 1000 metres. To save time and go through all the possible measurements by a calculator, I'm going to make a spreadsheet. I will change the length of the rectangle in steps of 10m. The formulas I will use in the spreadsheet are:
width = 500 - length
area = length * width
The results are as followed:
Using this results table I can draw a graph of area against height. From the graph, I can see that the maximum area of a rectangle with a range of lengths, ranging from 10m to 490min in 10m steps, is a square with a length of 250m, with a width of 250m and a perimeter of 1000m. The area of this square is 62500m². To work out the area of the rectangle more accurately, I will change the length of the rectangle in smaller steps of 1m. The results are as followed:
Using this results table I can draw another graph of area against height. From the graph, I can see that the maximum area of a rectangle is 62500m² exactly.
The shapes I have investigated suggest that:
- As you increase the number of sides you increase the area.
- For the given shape that has the maximum area, are regular shapes.
To investigate this further, the next shape I'm going to investigate is a regular pentagon.
Pentagon
Pentagons have 5 sides therefore I can divide it up into 5 isosceles triangles, and the angle at the top an isosceles triangle would be 360/5=72°.Therefore the other angles = (180-72)/2=54° each. I can split the isosceles triangle into 2 equal right-angled triangles and work out the area of the triangle, using trigonometry. I also know that each side of the pentagon = 1000/5=200m because it is a fifth of 1000m. I will use tangent to work out the height of the triangle because I know the adjacent (100m) and I want to find out the opposite.
height of triangle = 100m * tan54°
= 137.6m
Therefore, I can work out the area of the triangle because I know the height and base of it.
Area of triangle = base/2 * height
= 200m/2 * 137.6m
= 13760m²
Because I have worked out the area of a fifth of a pentagon, I have to multiply the area of the triangle by 5 because there are five triangles in the pentagon.
area of pentagon = area of triangle * 5
= 13760m² * 5
= 68800m²
Hexagon
Hexagons have 6 sides therefore I can divide it up into 6 isosceles triangles, and the angle at the top an isosceles triangle would be 360/6=60°. Therefore the other angles = (180-60)/2=60° each. I can split the isosceles triangle into 2 equal right-angled triangles and work out the area of the triangle, using trigonometry. I also know that each side of the hexagon is 1000/6=166.7m. I will use tangent to work out the height of the triangle because I know the adjacent (83.35m) and I want to find out the opposite.
height of triangle = 83.35m * tan60°
= 144.4m
Therefore, I can work out the area of the triangle because I know the height and base of it.
Area of triangle = base/2 * height
= 166.7m/2 * 144.4m
= 12035.74m²
Because I have worked out the area of a sixth of a hexagon, I have to multiply the area of the triangle by 6 because there are six triangles in the hexagon.
area of hexagon = area of triangle * 6
= 12035.74m² * 6
= 72214.44m²
Heptagon
Heptagons have 7 sides therefore I can divide it up into 7 isosceles triangles, and the angle at the top an isosceles triangle would be 360/7=51.4°. Therefore the other angles = (180-51.4)/2=64.3° each. I can split the isosceles triangle into 2 equal right-angled triangles and work out the area of the triangle, using trigonometry. I also know that each side of the heptagon is 1000/7=142.9m. I will use tangent to work out the height of the triangle because I know the adjacent (71.45m) and I want to find out the opposite.
height of triangle = 71.45m * tan64.3°
= 148.5m
Therefore, I can work out the area of the triangle because I know the height and base of it.
Area of triangle = base/2 * height
= 142.9m/2 * 148.5m
= 10610.3m²
Because I have worked out the area of a seventh of a heptagon, I have to multiply the area of the triangle by 7 because there are seven triangles in the heptagon.
area of heptagon = area of triangle * 7
= 13760m² * 7
= 96320m²
Octagon
Octagons have 8 sides therefore I can divide it up into 8 isosceles triangles, and the angle at the top an isosceles triangle would be 360/8=45°. Therefore the other angles = (180-45)/2=67.5° each. I can split the isosceles triangle into 2 equal right-angled triangles and work out the area of the triangle, using trigonometry. I also know that each side of the octagon is 1000/8=125m. I will use tangent to work out the height of the triangle because I know the adjacent (62.5m) and I want to find out the opposite.
height of triangle = 62.5m * tan67.5°
= 150.9m
Therefore, I can work out the area of the triangle because I know the height and base of it.
Area of triangle = base/2 * height
= 125m/2 * 150.9m
= 18862.5m²
Because I have worked out the area of a eighth of an octagon, I have to multiply the area of the triangle by 8 because there are eight triangles in the octagon.
area of octagon = area of triangle * 8
= 18862.5m² * 8
= 150900m²
General Equation of a polygon for n-sides
n = number of sides
h = height
a = area
Height of triangle of an n-sided shape
I will use tangent to work out the height of the triangle because I know the adjacent (71.45m) and I want to find out the opposite.
h = (500/n) * tan((90-(180/n))
Area of triangle of an n-sided shape
a = b/2 * h
a = (500/n) * ((500/n) * tan((90-(180/n)))
Area of an n-sided shape
a = n * area of triangle of an n-sided shape
a = n * ((500/n) * ((500/n) * tan(90-(180/n)))
Simplified:
a = 500 * ((500/n) * tan(90-(180/n)))
Multiplied out brackets:
a = 250000/n * tan(90-(180/n)))
I used the formulas above to see what happens as you increase the number of sides. The results are as followed:
From this results table, I can produce a graph of area against number of sides. From the graph and table, I can see that as you increase the number of sides, the area increase. Therefore, the shape that has the maximum area, with a perimeter of 1000m, is a circle because it has infinite sides. As n increases and approaches infinite, the shape should become a circle. Now I am going to investigate a circle and see if this is true.
Circle
a = area of circle
c = circumference of circle
r = radius of circle
a = r²
c = 2r
If the circumference was 1000m:
1000m = 2r
1000/2 = r
Simplified:
500/ = r
Substitute r in the formula to work out the area of a circle:
a = *(500/)²
a = *(500/)*(500/)
a = *(250000/²)
Simplified:
a = 250000/
a = 79577.47m²
The area of the circle is larger than the 30 side shape because a circle has infinite sides.
Conclusion
For any given n-sided shape, regular shapes have the maximum area and as the number of sides of a shape increase, the area increases. A circle has the maximum area because it has infinite sides. Therefore the farmer should fence off a circle with a perimeter 1000m because it would create an area of 79577.47m².
