1,000m of fence - what is the maximum area I can enclose?

The perimeter of an equilateral must equal twice the length and the width added together.

P=2(L+W)=1000 … (L+W=500)

Area = Length x Width

A = LxW → A=LW

There are many rectangles that have the rule that L+W=500 because there are decimals in between each full number (e.g. 500.1, 500.2)

Now that the above has been established, I will explore possible areas of the rectangles.

Below I have compiled my own line graph, but I know I can make a more accurate one using ICT, so I have included a line graph compiled using ICT.

By looking at the line graphs, I can tell that continuing would make a mirror of the curve.

Triangle:

There are different types of triangles to use – Right Angle, Isoceles, Equilateral, Scalene, Acute and Obtuse. Although triangles are the easiest shape to use, it may not be easy to work out area – because the height is not always specified.

Finding the Area – if the perimeter is 1,000m

Equilateral

To find the length of side x, I will have to use Pythagoras Theorem. The rule for Pythagoras theorem is a2 + b2 = c2.

a2 + b2 = c2

x2 + 166.6662 = 333.3332

x2 + 27777.77    = 111111.11

                                    83333.34 + 27777.77 = 111111.11

x = 288.6751461

x = 288.68

Now I know the height and base measurements, I will now continue to find the area of the triangle.

A = ½ bh

                                                                  = ½ (333.333 x 288.6751288)
                                                                 = ½ 96224.94671

                                                                  = 48112.47335m2

                                                                  = 48112.48m2

Right Angled

A = ½ bh

                                                                  = ½ 300x200
                                                                 = ½ 60000

                                                                  =  30000m2

                                                               

From these triangles, we can see equilateral triangles have a bigger area than right angled triangles – but still not as big as the square. I will further explore triangles in the forms of other shapes, such as hexagons and pentagons

Regular Pentagons

How I drew my (regular) Pentagons

1. I drew a circle, roughly to the size you wanted my pentagon to be
2. I then marked the centre of the circle
3. To get the sides, I divided 360 (full circle) by the amount of sides – in this case, 5. Each angle was be 72
4. I took a protractor and measured 72, marking the angles.
5. I connected the markings to generate the shape

Finding the Area

Finding the area of a pentagon is relatively simple. All I need to do is find out the area of one of the five triangles used to build the shape then multiply the answer by five.

From the diagram, I know the base – but I don’t know the two sides, so I can’t use Pythagoras theorem. I have the base and angles, so I can use trigonometry.

   

Tan  54 = . x .

                100

                                                                  1.3763819204711735382072095819109= . x .

                                                                                                                                      100

                                                        1.3763819204711735382072095819109 x 100 = x

                                                          137.63819204711735382072095819109 = x

                                                                                 137.64m = x (1dp)

                                         area of triangle = ½ bh

                                                                  = ½ 200 x 137.64
                                                                 = 0.5 x 200 x 137.64

                                                                  = 100 x 137.64

                                                                  = 13764

                                Pentagon = 5 x Area of triangle

                                                = 5 x 13764

                                                   = 68820m2

Now that I have sampled regular Pentagons, I will now move on to irregular pentagons.

Irregular Pentagons

                Irregular pentagons still have the same amount of sides as a    regular pentagon, but usually a different area.

           To find out the area of the irregular pentagon on the left, I will

           Divide this into two shapes – an isosceles triangle and a square. I will then find the areas of both shapes and add them together.

Area of triangle

To find the length of side x, I will have to use Pythagoras Theorem. The rule for Pythagoras theorem is a2 + c2 = b2.

                  a2 + c2 = b2

         1002 + c2 = 1502

    10000 + c2  = 22500

c = sqrt 12500

c = 111.80

Now I know the height and base measurements, I will now continue to find the area of the triangle.

A = ½ bh

            = ½ (200x111.80)
= ½ 22360

   = 11180m^2

Now I need to work out the area of the square.

                                                     A = LW

     A = 200 x 250

A = 50000

So, the area of the whole shape:

A of Irregular Pentagon = 50000 + 11180

                                      = 61180m2

Hexagon