This is not a totally authentic prediction yet, but as I go on with the investigation, I can add more to the hypothesis.
Investigating Triangles
Isosceles Triangles:
Many isosceles triangles with a perimeter of 1000m can be made. In order to calculate the area of all these triangles, a spreadsheet can be used.
The values of the base were entered, and from that, the lengths of the other two sides, the height, and then the area can be calculated.
To find the length of one side, we can simply minus the base from 1000m, and divide by two. In the spreadsheet, this formula is used =(1000-A2)/2
To then find the height, Pythagoras’ Theorem can be used. c2 = a2 + b2, can be re arranged to a = c - b , where a is the height, b is the base divided by two, and c is the length of one side. In the spreadsheet, this formula is used: =SQRT(POWER(B2;2)-POWER(A2/2;2)
Finally, to find the area, we simply multiply the base by the height, and divide that by two. For the spreadsheet, the cell containing the base value is multiplied by the cell containing the height value. The formula used is: =(A2*C2)/2
The length of the base cannot go up to 500m,
because that would leave 250m for the other
two sides , therefore making it an ‘impossible triangle’,
as the two con
The results from the spreadsheet can be shown on a graph:
The formula to work out the area of any Isosceles triangle, by simply knowing the length of the base(x), can then be:
area = ½ base x height
a = ½ x ( c - b )
a = ½ x 1000-x - x
2 2
Equilateral Triangles:
There can only be one equilateral triangle with a perimeter of 1000 metres. Each one of the sides will have a value of 1000m divided by three (333.3m).
To find the area, we can use the same equation:
a = ½ b h
= ½ 333.3 x ( c - b )
= ½ 333.3 x 288.675
= 48,112.50 m2
We can now see that the triangle with the largest
area is the Equilateral trianngle, and not the Isoceles.
Scalene Triangles:
After looking into Isosceles triangles, and the equilateral triangle, I do not see it necessary to carry on any further with scalene triangles.
I noticed that as the sides of the triangle became closer in length, the area increased, and therefore it makes sense that the triangle with equal sides has the maximum area.
Investigating Rectangles
To find out the areas of the rectangles with a perimeter of 1000m, I used a spreadsheet again. By filling in the values for the width, the length and the area of each triangle can be worked out.
We know that the perimeter of the rectangle will be 1000m, and that one length plus one width will be 500m, all the time.
To find the length, we minus the width value from 500m. The formula used in the spreadsheet is:
=500-A2
From that, the area can easily be worked out, using a =wl, and on the spreadsheet using:
=B2*A2
Again, my results from the spreadsheet can be shown on a graph:
From the previous formulae I used in the spreadsheet, I can form a new equation, which can be used to work out the area of any rectangle with a perimeter of 1000 metres, by simply knowing one width(x):
area = width × length
a = x (500-x)
a = 500x - x2
Extension:
Now that I have this formula, as an extension, I could have used Differentiation to find the rectangle with the maximum area.
Firstly, a value for either x or y must be substituted into the equation for area, i.e.
Area = xy
Perimeter = 2x + 2y
therefore -
x + y = 500m
y = 500 - x
Substitute y into the equation a = xy:
a = x ( 500- x )
a = 500x - x2
Now, I can differentiate:
dy = 500 - 2x = 0 it is equal to zero, because one of the rules of differentiation
dx states so.
2x = 500
x = 250 x and y are both equal to 250 metres, which is the correct
y = 250 answer.
Further Prediction:
After investigating triangles and rectangles, I now have more evidence to support my prediction. So far, I can see that the square has a larger area than the triangle, as I have said in my prediction.
Next, I can predict further and expect the pentagon to have a larger area than the square, and for all the areas of further shapes with more sides, to increase, until I reach the circle.
º
Investigating Polygons:
I have seen that the closer the sides in the shape are, the larger the area, so when investigating Polygons, I will only take the regular shapes.
Investigating Pentagons
In order to work out the lengths of each side of a regular pentagon, with a perimeter of 1000 metres, I must simply divide 1000m by five, since there are five equal sides.
To find the area, the pentagon is split into smaller triangles, and after calculating the area of one of these triangles, the whole area can easily be worked out.
We can use trigonometry to find the missing height.
We know the length of the opposite side, and we know theta, so we use TAN to find the adjacent.
Area of triangle = ½ b x h
a = ½ x 100 x 100 After finding the area of one triangle, I must
TAN 36o multiply that figure by the number of triangles
a = 6,881.91 m2 found in the pentagon.
Area of Pentagon = no. of triangles x area of 1 triangle
a = 10 x 6881.91
a = 68,819.1 m2
Investigating Hexagons
I will use the same method for finding the area of a regular hexagon, as I did for finding the area of a regular pentagon.
Area of triangle = ½ b x h
a = ½ x 83.33 x 83.33
TAN 30o
a = 6,013.58 m2
Area of Hexagon = no. of triangles x area of 1 triangle
a = 12 x 6013.58
a = 72,163.01 m2
Investigating Octagons
Again, the same method is used.
Area of triangle = ½ b x h
a = ½ x 62.5 x 62.5
TAN 22.5o
a = 4715.26 m2
Area of Octagon = no. of triangles x area of 1 triangle
a = 16 x 4715.26
a = 75,444.17 m2
Throughout investigating these polygons, I have been using the same formula over and over again. I have noticed that within the formula, the one thing that kept changing was the number of triangles in the shape ( t ). From that, I can form this equation to work out the area of any polygon:
a = t ( ½ b x b/2 )
TAN 0
Investigating Circles
There can only be one circle with a perimeter of 1000 metres. To find the area of this circle, we first need to know the radius.
Radius = 1000
2
r = 159.15
Area = r2
a = 3.142 x 159.152
a = 79,572.53
Conclusion
My prediction is correct, and it has been proven. There is a definite relationship between the number of sides and the area, as this graph shows:
