Area = 275m x 225m
Area = 61875
A square with four sides of 250m
Area = 250m x 250m
Area = 62500
After increasing the width by 25m and decreasing the width by 25m I have noticed that the square is the biggest 4 sided shape I will prove this by working out the area f the rectangle with a length of 451m and a width of 249m
Area = length x width
Area = 451m x 449m
Area= 62499
The area of the rectangle 451m by 449m is smaller than the area of the square 250m x 250m. This graph enforces this point.
Triangles
To work out the height we split the triangle in half so that we have a right angled triangle. We then use Pythagoras’ theorem for working out the area of a right-angled triangle. Pythagoras’ theorem for working out the area of triangles is
a ² + b ² = hypotenuse (c². We do not know b ² (which is the height) so we change the equation to make it so that we find b² and the equation becomes
c ² - a ² = b ²
This works out the height we then use the equation for working out height, which is:
½ base x height
These are isosceles triangles:
Base= 200m
Sides= 400m
Height² = 400² - 200²/2
= 160 000 - 100²
= 160 000 – 10 000
= 150000
Height = √150000
Height =387.29833462074168851792653997824
Area = ½ base times height
Area = 100 x 387.29833462074168851792653997824
Area to 1dp = 38729.8
base = 300m
sides = 350m
Height² = 350² - 150²
= 122500 – 22500
= 100000
Height = √100000
= 316.22776601683793319988935444327
Area = 150 x 316.22776601683793319988935444327
Area to 1dp = 47434.2
Base = 400m
Sides = 300m
Height² = 300 ² - 200²
Height = 90000 – 40000
Height = 50000
Height = √50000
Height = 223.60679774997896964091736687313
Area = 200 x 223.60679774997896964091736687313
Area to 1dp = 44721.4
A triangle with the base of 400 is smaller than a triangle with the base of 300 so I will work out the area of a triangle with te base of 350
Base = 350
Sides = 325
Height² = 325² – 17²
= 105625 – 30625
=75000
=√75000
height =273.8612787525830567284848914004
area = 175 x 273.8612787525830567284848914004
area to 1dp = 47925.7
Using the knowledge that a square was the biggest 4 sided shape I am going to work out the area of an equalateral triangle
Base = 333.333
Sides = 333.333
Area = ½ base x height
Height² = 333.3² - 166.7²
= 111110.888889 - 27788.89
=83321.998889
=√83321.998889
= 288.65550209375881564039836747977
Area = 166.7 x 288.65550209375881564039836747977
Area = 48118.9
To prove this is the biggest I can show you the area of a triangle with the base of 333.332 and the area of a triangle with the base of 333.334
Base = 333.332
Sides= 333.334
Height² = 333.334² – 166.666²
= 111111.555556 – 27777.555556
= 83334
Height = √83334
=288.67628929304186966690302476163
Area = 166.666 x 88.67628929304186966690302476163
Area to 1dp = 48112.5
Base = 333.334
Sides = 333.332
Height² = 333.332² -166.667²
= 111110.222224 - 27777.888889
= 83332.333335
height = √83332.333335
= 288.6734025416958984440235224874
area = 166.667 x 288.6734025416958984440235224874
area to 1dp = 48112.3
This graph makes my results clearer
I have come to the conclusion that when the sides are the same length the area is biggest I will use this in all my other investigations.
I am now going to see if increasing the number of sides makes a difference to the area.
To work out the area, I split the shapes into triangles and worked out the area of one triangle. To work out the area of the triangle I split it in half so that I have a right angled triangle I work out the height using sine cosine and tan times it by half the base to work out the area then multiply it by the number of sides the shape has.
Pentagon
The pentagon has 5 sides and therefore each side is 1000/5. Which means each side are 200m. The angle at the top of the triangle is 360 divided by 5. Each angle is 72˚.
Height =?
Tan 72/2 =200/2/?
Tan36 = 200/?
? = 100/tan 72
? = 137.6
Area = 5(137.6 x 100)
Area = 5 x 13760
Area = 68800
Hexagon
Base = 1000/6
= 166.666
Angle = 360 / 6
= 60
Tan 60/2 = 166.666/?
= Tan 30= 83.333/?
? = 83.333/Tan 30
? = 144.338
Area of triangle = 144.338 x 83.333
= 12028 x 6
Area of hexagon = 72168
Heptagon
Base = 1000/7
= 142.9
Angle = 360 /7
= 51.4
Tan 51.4/2 =142.9/2/?
Tan 25.7=71.4/?
? = 71.4/tan 25.7
? = 148.4
Area of triangle = 148.4 x 71.4
= 10595.76
Area of heptagon = 10595.76 x 7
= 74170.32
Decagon
Base = 1000/10
= 100
Angle = 360/10
= 36
Tan 36/2 = 100/2/?
Tan18 = 50/?
? = 50/tan18
? =153.9
Area of triangle = 153.9 x 50
=7695
Area of decagon = 7695 x 10
= 76950
Circle
The equation for working out the area of a circle is: -
Circumference = π x diameter²
Circumference = 1000m
1000m/π = diameter (d)
D = 318.3
Radius(r) = 318.3/2
R = 159.15
Area = π r ²
Area = π x 159.52
Area = 791572.5
Evaluation
My theory
My hypothesis was correct and this is now my theory , as the number of sides increases the area of the shape increases
If I repeated my investigation I could improve by investigating further and getting more results to prove my theory.
