Fencing - maths coursework

One side of a triangle = less than 500m

The following triangles that I investigation will be less than 500m per side.

Perimeter = 1000m

400m

100m

500m

Example:

Idea 1: S= 0.5 (a + b + c)

Idea 2: S= a + b + c

2

Idea 3: S= 1/2 (a + b + c)

Hero's formula = 500(500-500)(500-100)(500-400)

= 0

I will investigate equilateral, isosceles and scalene triangles:

200m

310m

490m

460m

S = 500 x 1 x 300 x 290

340m

Area = 16881.94302m²

200m

S = 500 x 300 x 40 x 160

Area = 30983.86677cm2

390m

200m

410m

S = 500 x 300 x 90 x 110

200m

Area = 38535.69774cm2

480m

320m

S = 500 x 300 x 180 x 20

Area = 23237.90008cm2

As I am getting closer to a scalene triangle, the bigger the area. So now I am going to investigate isosceles triangles

The formula I have used to find out the area of the scalene triangle is

=SQRT(D2*(D2-A2)*(D2-B2)*(D2-C2))

495m 495m

10m

S= 0.5 (495m + 495m + 10m) = 500m

500(500 - 490= 3)(500 - 5= 3)(500 - 5= 3)

= 2474.87m

377.5m 377.5m

245m

S= 0.5 (377.5m + 377.5m + 245m) = 500m

330m 330m

500 x 122.5 x 122.5 x 255

=4374.12m2

340m

S= 0.5 (330m + 330m + 340m) = 500m

500 x 160 x 170 x 170

= 48083.26m2

450m 450m

100m

S= 0.5 (450m + 450m + 100m) = 500m

500 x 50 x 50 x 400

=22360.68m2

450m 450m

100m

S= 0.5 (450m + 450m + 100m) = 500m

500 x 50 x 50 x 400 250.5m 250.5m

=22360.68m2

499m

S= 0.5 (499m + 250.5m + 250.5m) = 500m

400m 400m 500 x 1 x 249.5 x 249.5 =5578.99m2

200m

S= 0.5 (499m + 250.5m + 250.5m) = 500m

500 x 1 x 249.5 x 249.5

=5578.99m2

The formula to find out the area for the isosceles triangle is the same as a scalene triangle.

=SQRT(D2*(D2-A2)*(D2-B2)*(D2-C2))

I can see that the lengths of the sides get closer to each other the area increases. It looks like the greatest area will be when all the sides are exactly equal just like it was for the four-sided shapes. I will now test the equilateral triangle to check if my prediction is right.

000

3 333.33333333333m

H 333.33333333333m

166.6666667m

333.333333333332 - 166.66666672 = 83333.33336

= 83333.33336 = 288.6751346m

Area of a equilateral triangle = 0.5 x 288.6751346m x 333.33333333333m

= 48112.52243m2

Base

Hypnoses

Height

0.5 x height x base

Area

Isosceles

highest area

350

325

273.8162788

273.8162788 x 0.5 x 350

47917.84879m2

Scalene

highest area

350

250

400

875000000

43301.27019cm2

Equilateral

area

333.33333m

333.333333m

333.333333m

0.5 x 288.6751346m x 333.333333m

48112.52243m2

This shows that out of the triangle family equilateral triangle has the furthermost area.

Area = 0.5 x 288.6751346m x 333.33333333333m

= 48112.52243m2

333.33333333333m 250m

=

Area = 48112.52243m2 Area = 62500 m2

Perimeter = 1000m

Which one has the greatest area?

The square has the greatest area = 62500 m

The difference area between the square and the triangle was by = 62500 m - 48112.52243m

= 14387.477757m2

The square is greater because it has more sides.

These shapes are both regular polygons and the two shapes have given us the largest area from their family groups.

A I have discovered that the more sides you have the more the area will be so now I am going to investigate will different regular sharps such as: pentagons,

heptagons,

octagons and

decagons.

1000

5 = 200m

360

5 = 72 o

72 o 36 o

h

54 o

200m 100

90o + 36o = 126o

180o - 126o = 54o tan 54o = 100

h

h = 100 x tan 54o

h = 137.638192m

Area of the triangle: 100m x 137.638192m

2

=13763.8192m2

Area of the pentagon: 10 x 13763.8192m

= 68819.09602m2

Because we haled the length of the side (not aright angle) so we double the triangles.

1000

7 =142.8571429m

(Not drawn to scale)

360

7 = 51.42857143O 25.71428571O

51.42857143O

h

64.28571429o

(Not drawn to scale) 142.8571429m 71.42837143m

90o + 25.71428571o = 115.7142857o

180o - 115.7142857o = 64.28571429o

tan 64.28571429o = 71.42837143

h

h = 71.42837143m x tan 64.28571429o

h = 148.322957m

Area of the triangle: 142.8571429m x 148.322957m

2

=10594.4693m2

Area of the pentagon: 7 x 10594.4693m

= 74161.4785m2

Because we doubled the length of the side (not aright angle) so we halved the triangles.

1000

8 = 125m

360

8 = 45o

45o 22.5o

h

67.5 o

125m 62.5m

90o + 22.5o = 112.5o

180o - 112.5o = 67.5o tan 67.5o = 62.5

h

h = 62.5 x tan 67.5o

h = 150.8883476m

Area of the triangle: 62.5m x 150.8883476m

2

= 4715.260863m2

Area of the pentagon: 16 x 4715.360863m

= 75444.1738m2

Because we double the length of the side (not aright angle) so we halved the triangles.

1000

10 = 100m

(Not drawn to scale)

360

10 36o

36o 18o

72o

(Not drawn to scale) 100m 50m

90o + 18o = 108o

180o - 108o = 72o tan 72o = 50

h

h = 50 x tan 72o

h = 153.8841769m

Area of the triangle: 50m x 153.8841769m

2

=3847.104423m2

Area of the pentagon: 20 x 13763.8192m

= 76942.08843m2

Because we haled the length of the side (not aright angle) so we double the triangles.72o

I am now going to try and find a formula so I can find out 'n' number of sides.

The 'n' number of sides stands for 'any' number of sides.

(Not drawn to scale)

Here is a heptagon that I am going to use as an example.

1000m

n 'n' being the number of sides. In this case it will be '7'

360°

n As it is a regular polygon so there would be 'n' number of triangles. In this case there would be 7 as it is a heptagon that is why we would have to divide 360 by 7 to work out the degrees of one triangle.

Now I am going to split up the triangle in halve so I get a right angle triangle to find out the height.

180°

n

This would now be 180° so I have halved the triangle. (As I have said above)

500m this would be 500m because we have halved the triangle so the

n base will be halved as well.

180°

n

I will be using tan for this type of trigonometry as I have the opposite (500m) and I have the adjacent (180/n)

500 The formula for Tan ...°= Opposite

Adjacent

We have the angle 180°

n

x

This is the adjacent that we are This is the opposite that we already have.

trying to find out the height.

500

This would be the start of the formula to find 'n' number of sides

.

500

This is the length of the base of the triangle.

n

Tan 180

This is the degrees of one angle in the triangle.

n

The next part of the formula is:

1/2 x 1000 x 500

n

n

Tan 180

N

This final part of the formula is to put it all in order and make it simpler.

1/2 x 1000 x 500

n n

Tan 180

n

To prove that this is the formula I am now going to test it on one of the polygons.

I am going to prove the formula on the seven-sided shape.

1/2 x 1000 x 500

7 7

This is the first step to do.

(Add the '7's' in.)

Tan 180

7

71.42857143 500

7

Next work out this bit of the formula.

Tan 180

7

71.42857143

71.42857143

Next is to work out the two brackets.

0.4815746188

71.42857143 148.3229569 N ext work OR 71.42857143 X 148.3229569

this brackets

. 10594.49692 However, this is not the final bit of the formula, because there was

7 triangles in the heptagon so we have to times our answer (10594.49692) by 7 to get the area of the whole heptagon.

So finally the last step to find out the area of the whole polygon will be to:

0594.49692 X 7 = 74161.4785m2

I am going to prove the formula on the ten-sided shape just to make sure that my first try with the formula was not wrong.

1/2 x 1000 x 500

10 10

This is the first step to do.

(Add the '10's' in.)

Tan 180

10

50 500

10

Next work out this bit of the formula.

Tan 180

10

50

50

Next is to work out the two brackets.

0.324919696

50 153.884177 N ext work OR 50 x 153.884177

this brackets

. 7694.208848 However, this is not the final bit of the formula, because there was

10 triangles in the decagon so we have to times our answer (7694.208848) by 10 to get the area of the whole decagon.

So finally the last step to find out the area of the whole polygon will be to:

7694.208848 x 10 = 76942.08848m2

Now I can say thing that the formula I have found out does work as 1 have tested it out on two different types of shapes!

I am now going to test my formula in big numbers like: 100, 1,000, 10,000, 100,000, 100,000, and 1,000,00 etc....

00 sides

1/2 x 1000 x 500 5 5

100 100

0.031426266

Tan 180

100

100 x 795.5128999 795.518999 5 x 159.10258

79551.28999 m2

To make it easier I am just going to show you a table I had done in excel with all the areas of the sides I have said above but with a lot more detail.

I am now going to do three different graphics to show how the area of the polygons will be.

This is a graph showing the area that I have done but all of the regular polygons in one graph.

This is another graph show the different types or areas against the number of sides

This is a graph with all my results (the areas I have found.)

My graph and table clearly shows that as the numbers of sides increase also the area increases. So to find out the maximum area for a farmer with perimeter of 1000m I will need to look at the regular polygon with as many sides as possible. 100 sides would begin to look like a circle and as the number of sides increased it would get more and more like a circle so I will test the area of a circle to see if it has a bigger area than I have already found.

As I have notice when I was doing this investigation that the area in between the three- sides shape and the four-sided shape was: 62500m2 - 48112.52243m2 = 14387.47757. But when the sided of a polygon starts to increase the area in between the two polygons starts to decrease, like the difference between 8 sides and 10 sides: 76942.08843m2 - 75444.1738 = 14979.1463m2. You can see the difference between the area went even the sides 3, 4, 8, 10 sides are close to ether. If you think about it then how would it be if you try and find out the difference between 10,000 sides and 100,000? 79577.4715m2 - 79577.46893m2 = 0.00257. This now tells you that when the sided of a polygon increase the area will decrease as they is not much difference in the areas.

Here is some research that I have done to show to prove that when the sides of a polygon increases it looks more and more like a circle.

Firstly I am going to show you an 18-sided shape (octadecagon).

Now I am going to show you a 30-sided shape (triacontagon).

The website I have found this information is: http://en.wikipedia.org/wiki/Polygon#Names_and_types

Now I am going to be explaining to you how a tringle will find into a octagon and then a circle.

If you take the middle od a tringle and you pull each side outwards the corners of the triangle will be pulled inward a bit and the area inside the triangle will increase as you have maked the shape bigger, this will now look more and more like a octagon.

If you keep on repeating this ther shape of the polgyon will turn into a bigger polgyon eg: 12 sided shape and the slowly it will start to look more and more like a circle.

As you can see in the digram a lot of spcae is wasted from the triangle and the circle. However not much space is wasted from the octagon to the cricle, also when you have a triacontagon(30-sided shape) and you fit it in a circle there is hardly any wasted spce so this is tell us that, as the sided of the polgyen increases there is less wasted spcae.

Perimeter = 1000m

Perimeter = Circumference

Radius C = 2 ?r²

1000 = 2 ?r²

1000 - 2 = ?r²

500 = ?r²

500 - ? = r²

r = 159.15499431m

Area of a circle = ?r2

= x 159.154994312

= x 25330.29591

= 79577.47155m2

Finally I have discovered that a circle has the maximum area that the farmer will be able to build his fence over. A circle gives the most area with 1000 metres of fence.

Maths Fencing Coursework Rahul Malde 11e