To find the height I used Pythagoras theorem:
H=
X²- (500-X) ²
H=
X² – (250000-1000X+X²)
This is the simplified equation for the height which I have to use to achieve the area.
H= 1000X- 250000
I achieved the maximum area for the isosceles triangle when the side lengths were close to the base length. This is the same pattern that occurred with the rectangles occurred with the triangles as well. This means that it will occur with any regular polygon, the closer together the side lengths, the greater the area.
I have completed my research on isosceles triangles and I am going to move on to scalene triangles. For me to get the results as accurate as possible I will be using the hero’s formula.
A B
C
A+B+C are the three sides of the scalene triangle that add up to give 1000m. To find the area of this triangle I will need the Hero’s formula that gives me the area. A Scalene triangle has three different sides and angles.
A+B+C This gives you the S in the formula so the S will be 1000= 500 so S= 500
2 2
Hero’s formula:
S (s-a) (s-b) (s-c)
The maximum area the scalene triangle with a perimeter gives is when all three side lengths are close together. As the scalene triangle has side lengths that are all different lengths it will be harder for me to generate 3 numbers that are close to each other and add up to 1000. I then used hero’s formula which is needed to find the area of the scalene triangle.
The graph shows the side lengths in relation to the area of the scalene. The graph shows that as the side lengths get closer together, the area increases. I have completed my research on scalene triangles; I will start a new investigation on pentagons. In this investigation I am going to be using trigonometry, to find the area of the whole pentagon, I am going to split it up into 5 smaller triangles. The perimeter of the pentagon will once again be 1000m.
The area of the whole pentagon is 5T.
The Circle in the middle of the pentagon represents all of
360º of the angles in the middle of the pentagon.
To figure out the value of the angles in the middle I have to
divide 360 by 5 which gives me 72. This means that the
angle in the middle for one of the triangles is 72º.
Now that I have the angle for the triangle I have to work out the base length of the triangle. To work this out I have to do 1000 = 200m
5
This is a section of the pentagon in the form of a triangle.
To find the area of the triangle I have to use the formula
BXH = 200XH
2 2
200
Now all I have to do is work out what the height is. To do this I am going to split the triangle into two right-angled triangles so that the angle halves leaving me with 36º and 100m as the base.
To work out the height I have to use trigonometry. Height = 100 = 137.64 (2 d.p)
Tan 36
BXH = 200X137.64 = 27528 = 13763.8
2 2 2
The area of the triangle is 13763.8m². To get the area of the whole pentagon I am
going to times the area of the triangle by 5. This gives me the whole area of the whole
pentagon which is 68819.1m².
Now that I have completed my study on pentagons and I will move on to hexagons. In this investigation I am going to using trigonometry to find the area of the whole shape. I am going to split up the hexagon into 6 smaller triangles so that I can find the area of 1 of the triangles and then times it by 6 to find the area of the whole hexagon. I will also have to keep in mind that the perimeter has to be 1000m.
The area for the whole hexagon is 6T
The in the middle has a total of 360º
One Section (T) 360 = 166.7
6
As the fencing has to 1000 metres I have to divide 1000 by 6 to give me length of the base with is equal to 166.7.
To find the height of the triangle I am going to have to cut the triangle in half so that it is a right-angled triangle. I need to make it a right-angled triangle so that I can use trigonometry to find out what the height is.
HEIGHT= 83.335 so height= 144.34
TAN30
Finally to get the area I have to use the formula BXH = 166.67 X 144.34 = 24057.15
2 2 2
= 12028.1m
The area 12028.1 is the area of the triangle I got from the whole hexagon, so to get the area of the whole hexagon I have to times 12028.1 by 6, this gives me 72168.7836. This means that the maximum area that a hexagon can cover with a perimeter of 1000m is 72168.8m².
I have completed my research into the hexagon and I will move on to heptagons. In this investigation I am going to use trigonometry to find the height of the triangles of which I am going to split the heptagon into to work out the area of the triangle.
The heptagon has 7 sides and I am going to be giving it
1000m perimeter. To find the base of the heptagon I am
going to divide 1000 by 7 to give me 142.86.
To find out the angle of one of the triangles split up from the heptagon, I am going to 360 = 51.43º.
7
Now all I have to find is the height which I am going to use trigonometry to find out. I am going to have to half all the measurements in the triangle so that it leaves me with a right-angle triangle. I need it to be a right-angle triangle because I can then use trigonometry on it to find out what the height is.
To find the height I have to use the formula tan∂ = Opp = 71.43
Adj Tan25.715 = 148.32
Now that I have the height and the base of the triangle I can now calculate the area of the triangle, and the times that by 7 to give me the area of the heptagon.
142.86 X 148.32 = 10594.5
2
Now to get the area of the whole triangle I am going to times 10594.5 by 7 to give me the area of the whole heptagon which is 74161.5m². I have mow completed my study on the heptagon and I am going to move on to plotting a table to compare the number of sides and area of the regular polygons I have researched.
The final shape I am going to do some research on is the circle. I am going to study the circle because a circle is an easy shape to work with and because I think that the largest area will come from the circle. I have noticed some patterns that come from the shapes I have studied, if I was to put all the shapes into a circle then as the number of sides increase so does the amount of space that they take up inside the circle. This means that only if I was to use a circle that has no side’s will I get the maximum value for 1000m of fencing.
The line that is shaded red is diameter of the circle and the
blue line represents the radius. The circumference is 2ПR.
The circumference is 1000m and the formula for working out the circumference is 2ПR.
The radius is 500. This is because if I re-arrange the formula to work out the circumference to make the radius the subject I get R = 1000
2П
Also if I put the radius of 500 into the formula 2ПR then I get 1000 as the circumference which is what I want.
Finally to find the area of the circle I am going to use the formula ПR². If I substitute the values for the corresponding letters I will get the area of the circle with a perimeter of 1000m.
ПR² = П X 500²
П X 250000
ПR² = 79577.4715
The area is 79577.4715m²; this shows that after searching many different shapes, the maximum area for a shape with a perimeter 1000m is found with the circle. I am now going to move on to working out a general formula, where the area can be calculated simultaneously for any sided shape and keeping in mind that it should have a perimeter on 1000m.
General Formula
I will now generate a formula to be used to work out the area of any sided regular polygon that has a perimeter 1000.
- The ‘n’ will represent the number of sides that the regular polygon has
- To find the length of just one side by dividing 1000 by ‘n’
This is the final and simplified general formula.
Area = 250000 + 1
n Tan(180)
n
The table below shows the general formula in use to find the area of shapes with any number of sides. I am only going to be showing the first 20 to show that the formula works.
The general formula limit shows that as you increase the number of sides, the area also increases. The area will increase less and less until it reaches a limit where it doesn’t matter how many more sides you add to the shape the area will not increase. A limit is something that you cannot pass or exceed. The graph shows that the area is nearing its limit, the limit is also the area of the circle and you cannot get any higher value then it. I have only showed you up to the 32 sided shape and if I continue my research I am sure that I will find the limit.
Radians
A radian is the measure for angles just as degrees is the measure for an angle.
To find the circumference you do 2ΠR
C= 2ΠR
360º = 2Π
180º = Π^c (c is the radiant that measures the angle)
90º = Π/2^c
45º = Π/4^c
60º = Π/3^c
30º = Π/6^c
The radian formula is quite the same as the general formula, this is because the radian is a more detailed figure of degrees, you use radians for more detailed answers because they are more accurate then degrees.
The radian formula would be:
250000/n X 1/(tan (П/n)
The only part changed about this formula compared to the general formula is that the radian formula has got П added to it.
Small Angle Theory
Sin O is roughly the same as theta O
Tan O is roughly the same as theta.
Now the radian formula is similar to the formula of the ‘n’ sided polygon.
Area = 500² X 1 tan(П/n)
n
As you make the ‘n’ larger the angle gets smaller, as it approaches 0 it makes the angle smaller.
As the angle gets smaller, the approximation will be much clearer and much more accurate.
Conclusion
I have now come to the end of my research; I have found out that the maximum area is achieved from a circle. All the research on the number of sides that give the maximum showed that the maximum area the circle with a perimeter 1000m is 79577.4715m². As the numbers of sides on a shape begin to increase, the shapes become more and more circular. So in conclusion the shape that gives you the highest area is the circle and the more sides you put on a shape the more circle like they become.
This is a square in a circle and the edges of the square are touching the circle. If I added more sides to the square the area filled by the shape would increase and would look more like a circle.
This is an octagon in a circle and all of the edges are touching the circle. As you can see the octagon is more circular looking and the space left between the octagon and the circle has decreased. This means that the more sides a shape has the more space they fill in a circle.
