table. Also another way of checking that my theory is right is by trying out the formula ½(base x height). When either the height or the base is a large number the area of the triangle will be larger. In the diagram below you can visually tell that when the height is at its highest possible point the area is at its largest.
You can see that the triangles all have the same base and by changing the perpendicular height affects the triangles area. I have found the area of all the shapes above below is a table of them.
The triangle with the largest area is the isosceles, now all I have to is find out which isosceles has the largest area in order to do this I will start with a base of 50m and find the area of that shape and then keep increasing the base by 50m and then find the area of that shape and so on. To represent this as a more understandable format I have changed the data into a scatter graph (below).
A graph to show how the side lengths of an isosceles triangle affect the area.
The graph is a parabola. As you can see the graph is at it highest point when the two sides are numerically near each other.
It is now proven that when the triangle is an isosceles the area is larger. So now all we have to do to find the triangle that has the largest area out of the isosceles. To find the area of the triangles I will have to use Pythagoras’ formula.
E.g.
h = height
h = 450^2 + 100^2
h = 202500 + 10000
h = 460.9772, this is then used in the
formula base x height
so, Area of triangle = 100 x 400.9772
= 46097.72m^2
I will use his method with the rest of the values. Below is a table showing the results of the values.
Now we can transform this into a graph, which is easier to read.
This graph is a parabola. It is symmetrical. The highest point of the graph is where the area is the largest. We cannot estimate the highest point but you can tell that it is between the values 300 and 350. I can use these values as margins for when I home in to find the real answer.
The triangle with the largest area has the lengths of 333.3, 333.3, and 333.3, which is a regular equilateral triangle.
Now all I have to do is find the area of this triangle and then I’m done.
Area of Triangle= sqrt (500(500-333.3)(500-
333.3)(500-333.3))
= SQRT 500 x 166.67 x166.67 x
166.67
=sqrt 2314953706.4815
I have done the three sided shape now I have to do the four sided, which is the parallelogram
Now I am going to investigate parallelograms because this is the next shape that has four sides after the three-sided triangles. In this section I have to find out the parallelogram with the perimeter of 1000 metres with the largest are just like the triangles.
E.g.
h
There is no need to draw all of them out, it would be easier to draw out a table and work out the area of each and record the answer. In any rectangle the two different sides have to have a total of 500m, because each side that is opposite has to be the same length. Therefore in a rectangle of 400 x 100, there are two sides of 100 and two sides of 400. This is what I have done on the next page.
This is for the rectangles.
Now I will put this in a graph to make it easier and more convenient to read.
A graph to show how the base affects the area
This graph is a parabola and is symmetrical, and like the triangles it has a highest point which we can only estimate so I will have to take rough values and then home in on them. So the rough values are 200 and 300.
You can clearly tell that the parallelogram with the lengths 250 x 250 has the largest area. It is a square; just like the triangle it is a regular shape. This enough to prove hat all the shape with different number of sides are regular. I can now continue with this investigation using just the regular shapes straight away instead of trying to find the certain shape with the largest area.
Now we are up to the stage where we know the largest area of the three and four-sided shape and now we need to do the five-sided regular shape, which is the regular pentagon.
As a regular pentagon he sides all have to be the same lengths, this would make them 1000/5 because there are five sides which is two hundred. There is no formula to find out the area of a pentagon so I will have to work out the area of this shape logically and I believe that the easiest way of approaching this problem is to split the pentagon up into five sections, these sections will be isosceles triangles, finding the area of one and multiplying it by the number of sections which could also be referred to as the number of sides. We now know the length of on side and we can now work out he angle of one section by dividing 360 by five, which gives us 72°.
This is one section from the pentagon. We can now work
B out the area by splitting it in half. By doing this the
angle is halved as well as the length.
Angle “a” is 54° because angles in a
Triangle add up to 180°.
A= 180-72/2
= 54°
A c every isosceles triangle can be split into two h
right-angled triangles and the area of one
triangle can be found by using trigonometry
Using SOHCAHTOA I can use the tangent to find out the height of
the split triangle.
H= 100
Tan 36
H= 137.638
Now I put this in the formula b x h
2
Area of split triangle= 100 x 137.638
2
Area of split triangle= 6881.910
I will now multiply the answer by 10 because I have found the area of half a triangle.
6881.910 x 10= 68819.1m^2
So far I have found out that as the area increases as the number of sides do. I am still going to investigate further with a regular hexagon, heptagon, octagon and nonagon. I am going to use the same method that I used to solve the area of the pentagon to solve the area of the Hexagon.
H= 83.3
Tan 30
H= 144.28
Area of X= 83.3 x 144.28
2
= 6009,262
Area of hexagon= 6009,262 x 12
=72111.144m^2
H
X
1000/7= 142.857m
360/7=51.4°
H= 71.4285
Tan 25.714
H=148.3247
Area of triangle= 71.4285 x 148.3247
2
Area of triangle= 5297.3054
Area of Heptagon= 5297.3054 x 14
= 74162.27584m^2
H= 62.5
Tan 22.5
H= 150.8
Area of triangle= 62.5 x 150.8
2
Area of triangle= 4712.5
Area of octagon= 4712.5 x 16
= 75400m^2
Now we have enough evidence to prove the statement that when the number of sides is increased, so is the area. Here is a results table of all the shapes and their largest area.
The relationship in this graph shows that as the number of sides increases so does the area.
A graph to show the elation between the number of sides and the area
Now my aim is to find a formula, which can be used to find out the area of the polygon.
n= number of sides .
What I have to find out:
FORMULA= xyn
x
y
x= 500
n__
tan(180)
n
y= 5oo
n
500 5oo
n___ x n n
tan180
n
. We could try this formula to see if it works by applying it to one of the polygons that we have already done. I am going to try the heptagon.
500^2
n tan(180)
n
Area= 250000
tan(180/7)
Area= 250000 x n
tan 25
Area= 250000 x n
3.264
Area= 74162.27584m^2
I can also use this formula to find the area of any other polygon with any sides. Although you cannot use it to find the area of a circle because we do not know how many sides it has because it is an infinite number. This is the answer to this investigation. The circle is the shape which has the largest area inside because it as an infinite number of sides.
To find the area you would have to use πr^2. We do not know what r is but you could find out by re- arranging the formula for the circumference.
1000= 2πr
500= πr
r=159.1549
∴ π x 159.1549 x 15901549 = area of a circle
