20m
480m
Then I used the formula;
Area of a rectangle = length x width
Area = 480m x 20m
Area of a rectangle = 9600 meters.
I added 10 each time to the width and worked out the length and the area and I put all my results on this chart.
I then plotted my results on a graph.
This graph represents the results that I found when I investigated the largest area of a square and a rectangle with a perimeter 1000m. The graph shows that the largest area of a square or rectangle is 62500m²,which is a square.
Therefore I have proved that the four-sided shape with the maximum area is a square, which is the only regular quadrilateral.
I have proved that the square has a greater area than any rectangle with the perimeter of 1000 meters. From these results, I believe that regular shapes have greater areas than irregular shapes.
I will now prove this theory by trying some other irregular quadrilaterals. Another quadrilateral is a Parallelogram.
Area = base x perpendicular height
450m
50m
I know that the hypotenuse is always bigger than the adjacent and the opposite sides and therefore I know that since 50 meters is the hypotenuse, the height has to be less than 50 meters. This is because the hypotenuse is always bigger than the other two sides in a right-angled triangle and therefore the height will be less than 50 meters. Since the area of a parallelogram = height x base, the height will be less than in a rectangle and therefore have a smaller area.
The same theory applies to a Trapezium, as they are made up of diagonals and the heights will be less than the sides that are made up of hypotenuses.
This is because the hypotenuse is the longest side in a triangle and this is why the height is always going to be smaller.
To work out the length of a trapezium the following formula is needed:
½(a + b) x h
a
B
Area of a trapezium = ½ (a + b) h
Kites and Rhombuses
I do not need to investigate these shapes, both because if I take an equilateral kite, it becomes a Rhombus and a Rhombus is a square. The square has the biggest area of all the quadrilaterals. This shows that you do not have to investigate Kites and Rhombuses as they all fall under the categories of the square.
Now I will investigate the areas of Triangles.
First I will try an equilateral triangle. It has all equal sides and the same angles.
There are three sides that all have to be the same length. The three lengths have to equal 1000 metres so you therefore have to divide 1000 by 3.
1000÷3 = 333.3333333 (to 7 d.p)
333.3333333m 333.3333333m
333.3333333m
To find the area of a triangle you have to use the formula;
Area of a triangle = ½ x base x height
The base is 333.3333333metres.
To find the height, we will divide the triangle into 2.
333.3333333 (hypotenuse) (c)
(a)
166.6666667
(b)
The base is half of 333.3333333 metres which;
Base = 333.3333333 ÷2
Base = 166.6666667 metres.
Now use Pythagoras’ Theorem,
It is a squared + b squared = c squared
C is always the hypotenuse, and the hypotenuse is always the angle opposite the angle
a ² + b ² = c ²
Find a
Therefore, rearrange the formula;
a ² = c ² – b ²
a ² = 333.3333333 ²- 166.6666667 ²
a ² = 111111.1111 – 27777.77779
a ² = 83333.33331
(Square route)
a = 288.6751346 metres.
Therefore,
The height is 288.6751346 metres.
The base is 333.3333333 metres.
Area of a triangle = ½ base x height.
Area = ½ x 333.3333333 x 288.6751346
Area = 48112.52243m²
The area of an equilateral of 1000 metres is 48112.52243m²
Now I am going to investigate an Isosceles Triangle:
Base = 50m
Side = 450m
Side = 450m
Height² (h)= a²+b²
H² = 495²- 10²
H² = 245025-100
H² = 244925
H = 489.8979
A= ½ base x height
A= ½ 20 x 498.8979
A= 4898.979m²
Area for an Isosceles triangle = 4898.979m²
I am not going to do right angled-triangles because they are scalene triangles and because they are irregular triangles.
Polygons
I am now going to investigate Pentagon: As there are 5sides in a pentagon, I am going to divide 1000 by 5, which equals 200m per side.
Pentagons can be divided into 5 equal Isosceles triangles.
Exterior angle = 360°/n or 360°/5 = 72°
Interior angle =180° – 72° =108°
Area = ½ base x height x number of triangles
200m
For this we will use trigonometry:
Tan = opposite/ adjacent
Tan 36= 100/ adjacent
Adjacent = 100/ Tan36
Adjacent = 137.64
Height = 137.638
Area = ½ 200 x 137.638
Area = 13763.8192m²
13763.8192m x 5 =
The Area of the Pentagon is 68819.096m²
I am now going to investigate the area of a hexagon:
Exterior Angle = 360°/6 =60°
Interior angle = 180° -60°= 120°
Tan30° = 83.33/H
H = 144.337m
Area = ½ base x height
Area = ½ 166 2/3 x 144.337
Area = 12028.083
12028.083 x 6 = 72168.65m²
Area of hexagon = 72168.65m²
From these two shapes I can conclude, that as the number of sides increase, so does the area.
Polygon Formula
Each shape’s perimeter = 1000m
Therefore one side = 1000/n
Regular polygons are split up into triangles.
The formula for the small triangle inside the regular polygon
=½ base x height
Then you just multiply by the number of sides (n) :
n (½ base x height)
The final formula = n (½ x 1000/n x height)
We then had to find the height of the triangle, using tan. I divided the triangle into 2. 360/n, then became 180/n, as we divided the triangle into 2.
Tan= opposite/ Adjacent
Tan (180/n) = opposite/ adjacent
Height = ½ x 1000/n/ Tan (180/n)
A = n(( ½ x 1000/n) x ½ x1000/n/Tan (180/n))
A= n(500/n x (500/n /Tan( 180/n)))
A= n(250000/n)/(n x tan (180/n))
A= n(250000/n²xTan (180/n)
Formula for an n sided shape with a 1000 perimeter is:
A= 250000/(n x Tan (180/n)
If we put the formula into practice the rest of the polygons:
Heptagon
Area = 250000/ (n x tan (180/n)
Area = 250000/ (7 x tan (180/7)
Area = 74161.478m²
Octagon
Area = 250000/ (n x tan (180/n)
Area = 250000/ (8 x tan (180/8)
Area = 75444.174m²
Nonagon
Area = 250000/ (n x tan (180/n)
Area = 250000/ (9 x tan (180/9)
Area = 76318.717m²
Decagon
Area = 250000/ (n x tan (180/n)
Area = 250000/ (10 x tan (180/10)
Area 76942.088m²
11- sided shape
Area = 250000/ (n x tan (180/n)
Area = 250000/ (11 x tan (180/11)
Area = 77401.983m²
Dodecahedron
Area = 250000/ (n x tan (180/n)
Area = 250000/ (12 x tan (180/12)
Area = 77751.058m²
20- sided shape
Area = 250000/ (n x tan (180/n)
Area = 250000/ (20 x tan (180/20)
Area = 78921.894m²
50 –sided shape
Area = 250000/ (n x tan (180/n)
Area = 250000/ (50 x tan (180/50)
Area = 79472.724m²
100- sided shape
Area = 250000/ (n x tan (180/n)
Area = 250000/ (100 x tan (180/100)
Area = 79551.290m²
150- sided shape
Area = 250000/ (n x tan (180/n)
Area = 250000/ (150 x tan (180/150)
Area = 79565.836m²
1000- sided shape
Area = 250000/ (n x tan (180/n)
Area = 250000/ (1000 x tan (180/1000)
Area = 79577.210m²
10000- sided shape
Area = 250000/ (n x tan (180/n)
Area = 250000/ (10000 x tan (180/10000)
Area = 79577.469m²
As the working out above shows, that the more sides there the larger the area is. But the increase is very small.
My final investigation is the Circle:
Perimeter = 2Πr
2Πr = 1000m
Πr = 1000 ÷ 2
Πr = 500m
r = 500 ÷ Π
r = 159.155
AREA = Πr²
= Π (159.155) ²
= Π (25330.296)
Area of a Circle = 79577.47155m ²
I have proven my hypothesis as the circle has the largest area with a perimeter of 1000m.
Radians
I am going to develop on the area of a circle as it is very similar to the area of a regular polygon which,
Area = 250000 ÷ (n x tan (180/n)
Radius = 1000 ÷ 2Π
From this I can see that I can get the formula of a regular polygon. This is how:
Place the above equation into a formula for a circle and you get
Area = Π x 500² / Π²
250000 / Π
Eventually the Polygon will turn into a circle, as a circle has an infinitve number of sides.
To develop on the idea of
250000/ n x tan (180/n) I am changing the formula into radians
Π / 180 is the method I found out for radians.
250000/nx tan (180/n) x (Π /180)
From this I can see that 180 and 180 cancel and as tan doesn’t make a difference in the equation so I am left with
25000/ n(Π/n)
The n’s cancel out…
250000/ Π
Which shows again that the formula for a circle and polygon is very similar
I can also see that I can make an equation to use any perimeter.
If I take my original formula:
A = n(( ½ x 1000/n) x ½ x1000/n/Tan (180/n))
A = n(( ½ x p/n) x ½ x p/n/Tan (180/n))
